Ideals in multivariate polynomial rings

Sage has a powerful system to compute with multivariate polynomial rings. Most algorithms dealing with these ideals are centered on the computation of Groebner bases. Sage mainly uses Singular to implement this functionality. Singular is widely regarded as the best open-source system for Groebner basis calculation in multivariate polynomial rings over fields.

EXAMPLES:

We compute a Groebner basis for some given ideal. The type returned by the groebner_basis method is PolynomialSequence, i.e., it is not a MPolynomialIdeal:

sage: x,y,z = QQ['x,y,z'].gens()
sage: I = ideal(x^5 + y^4 + z^3 - 1,  x^3 + y^3 + z^2 - 1)
sage: B = I.groebner_basis()
sage: type(B)
<class 'sage.rings.polynomial.multi_polynomial_sequence.PolynomialSequence_generic'>

>>> from sage.all import *
>>> x,y,z = QQ['x,y,z'].gens()
>>> I = ideal(x**Integer(5) + y**Integer(4) + z**Integer(3) - Integer(1),  x**Integer(3) + y**Integer(3) + z**Integer(2) - Integer(1))
>>> B = I.groebner_basis()
>>> type(B)
<class 'sage.rings.polynomial.multi_polynomial_sequence.PolynomialSequence_generic'>

Groebner bases can be used to solve the ideal membership problem:

sage: f,g,h = B
sage: (2*x*f + g).reduce(B)
0

sage: (2*x*f + g) in I
True

sage: (2*x*f + 2*z*h + y^3).reduce(B)
y^3

sage: (2*x*f + 2*z*h + y^3) in I
False

>>> from sage.all import *
>>> f,g,h = B
>>> (Integer(2)*x*f + g).reduce(B)
0

>>> (Integer(2)*x*f + g) in I
True

>>> (Integer(2)*x*f + Integer(2)*z*h + y**Integer(3)).reduce(B)
y^3

>>> (Integer(2)*x*f + Integer(2)*z*h + y**Integer(3)) in I
False

We compute a Groebner basis for Cyclic 6, which is a standard benchmark and test ideal.

sage: R.<x,y,z,t,u,v> = QQ['x,y,z,t,u,v']
sage: I = sage.rings.ideal.Cyclic(R,6)
sage: B = I.groebner_basis()
sage: len(B)
45

>>> from sage.all import *
>>> R = QQ['x,y,z,t,u,v']; (x, y, z, t, u, v,) = R._first_ngens(6)
>>> I = sage.rings.ideal.Cyclic(R,Integer(6))
>>> B = I.groebner_basis()
>>> len(B)
45

We compute in a quotient of a polynomial ring over \(\ZZ/17\ZZ\):

sage: R.<x,y> = ZZ[]
sage: S.<a,b> = R.quotient((x^2 + y^2, 17))
sage: S
Quotient of Multivariate Polynomial Ring in x, y over Integer Ring
by the ideal (x^2 + y^2, 17)

sage: a^2 + b^2 == 0
True
sage: a^3 - b^2
-a*b^2 - b^2

>>> from sage.all import *
>>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2)
>>> S = R.quotient((x**Integer(2) + y**Integer(2), Integer(17)), names=('a', 'b',)); (a, b,) = S._first_ngens(2)
>>> S
Quotient of Multivariate Polynomial Ring in x, y over Integer Ring
by the ideal (x^2 + y^2, 17)

>>> a**Integer(2) + b**Integer(2) == Integer(0)
True
>>> a**Integer(3) - b**Integer(2)
-a*b^2 - b^2

Note that the result of a computation is not necessarily reduced:

sage: (a+b)^17
a*b^16 + b^17
sage: S(17) == 0
True

>>> from sage.all import *
>>> (a+b)**Integer(17)
a*b^16 + b^17
>>> S(Integer(17)) == Integer(0)
True

Or we can work with \(\ZZ/17\ZZ\) directly:

sage: R.<x,y> = Zmod(17)[]
sage: S.<a,b> = R.quotient((x^2 + y^2,))
sage: S
Quotient of Multivariate Polynomial Ring in x, y over Ring of
integers modulo 17 by the ideal (x^2 + y^2)

sage: a^2 + b^2 == 0
True
sage: a^3 - b^2 == -a*b^2 - b^2 == 16*a*b^2 + 16*b^2
True
sage: (a+b)^17
a*b^16 + b^17
sage: S(17) == 0
True

>>> from sage.all import *
>>> R = Zmod(Integer(17))['x, y']; (x, y,) = R._first_ngens(2)
>>> S = R.quotient((x**Integer(2) + y**Integer(2),), names=('a', 'b',)); (a, b,) = S._first_ngens(2)
>>> S
Quotient of Multivariate Polynomial Ring in x, y over Ring of
integers modulo 17 by the ideal (x^2 + y^2)

>>> a**Integer(2) + b**Integer(2) == Integer(0)
True
>>> a**Integer(3) - b**Integer(2) == -a*b**Integer(2) - b**Integer(2) == Integer(16)*a*b**Integer(2) + Integer(16)*b**Integer(2)
True
>>> (a+b)**Integer(17)
a*b^16 + b^17
>>> S(Integer(17)) == Integer(0)
True

Working with a polynomial ring over \(\ZZ\):

sage: R.<x,y,z,w> = ZZ[]
sage: I = ideal(x^2 + y^2 - z^2 - w^2, x-y)
sage: J = I^2
sage: J.groebner_basis()
[4*y^4 - 4*y^2*z^2 + z^4 - 4*y^2*w^2 + 2*z^2*w^2 + w^4,
 2*x*y^2 - 2*y^3 - x*z^2 + y*z^2 - x*w^2 + y*w^2,
 x^2 - 2*x*y + y^2]

sage: y^2 - 2*x*y + x^2 in J
True
sage: 0 in J
True

>>> from sage.all import *
>>> R = ZZ['x, y, z, w']; (x, y, z, w,) = R._first_ngens(4)
>>> I = ideal(x**Integer(2) + y**Integer(2) - z**Integer(2) - w**Integer(2), x-y)
>>> J = I**Integer(2)
>>> J.groebner_basis()
[4*y^4 - 4*y^2*z^2 + z^4 - 4*y^2*w^2 + 2*z^2*w^2 + w^4,
 2*x*y^2 - 2*y^3 - x*z^2 + y*z^2 - x*w^2 + y*w^2,
 x^2 - 2*x*y + y^2]

>>> y**Integer(2) - Integer(2)*x*y + x**Integer(2) in J
True
>>> Integer(0) in J
True

We do a Groebner basis computation over a number field:

sage: K.<zeta> = CyclotomicField(3)
sage: R.<x,y,z> = K[]; R
Multivariate Polynomial Ring in x, y, z over Cyclotomic Field of order 3 and degree 2

sage: i = ideal(x - zeta*y + 1, x^3 - zeta*y^3); i
Ideal (x + (-zeta)*y + 1, x^3 + (-zeta)*y^3) of Multivariate
Polynomial Ring in x, y, z over Cyclotomic Field of order 3 and degree 2

sage: i.groebner_basis()
[y^3 + (2*zeta + 1)*y^2 + (zeta - 1)*y + (-1/3*zeta - 2/3), x + (-zeta)*y + 1]

sage: S = R.quotient(i); S
Quotient of Multivariate Polynomial Ring in x, y, z over
Cyclotomic Field of order 3 and degree 2 by the ideal (x +
(-zeta)*y + 1, x^3 + (-zeta)*y^3)

sage: S.0  - zeta*S.1
-1
sage: S.0^3 - zeta*S.1^3
0

>>> from sage.all import *
>>> K = CyclotomicField(Integer(3), names=('zeta',)); (zeta,) = K._first_ngens(1)
>>> R = K['x, y, z']; (x, y, z,) = R._first_ngens(3); R
Multivariate Polynomial Ring in x, y, z over Cyclotomic Field of order 3 and degree 2

>>> i = ideal(x - zeta*y + Integer(1), x**Integer(3) - zeta*y**Integer(3)); i
Ideal (x + (-zeta)*y + 1, x^3 + (-zeta)*y^3) of Multivariate
Polynomial Ring in x, y, z over Cyclotomic Field of order 3 and degree 2

>>> i.groebner_basis()
[y^3 + (2*zeta + 1)*y^2 + (zeta - 1)*y + (-1/3*zeta - 2/3), x + (-zeta)*y + 1]

>>> S = R.quotient(i); S
Quotient of Multivariate Polynomial Ring in x, y, z over
Cyclotomic Field of order 3 and degree 2 by the ideal (x +
(-zeta)*y + 1, x^3 + (-zeta)*y^3)

>>> S.gen(0)  - zeta*S.gen(1)
-1
>>> S.gen(0)**Integer(3) - zeta*S.gen(1)**Integer(3)
0

Two examples from the Mathematica documentation (done in Sage):

We compute a Groebner basis:

sage: R.<x,y> = PolynomialRing(QQ, order='lex')
sage: ideal(x^2 - 2*y^2, x*y - 3).groebner_basis()
[x - 2/3*y^3, y^4 - 9/2]

>>> from sage.all import *
>>> R = PolynomialRing(QQ, order='lex', names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> ideal(x**Integer(2) - Integer(2)*y**Integer(2), x*y - Integer(3)).groebner_basis()
[x - 2/3*y^3, y^4 - 9/2]

We show that three polynomials have no common root:

sage: R.<x,y> = QQ[]
sage: ideal(x+y, x^2 - 1, y^2 - 2*x).groebner_basis()
[1]

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> ideal(x+y, x**Integer(2) - Integer(1), y**Integer(2) - Integer(2)*x).groebner_basis()
[1]

The next example shows how we can use Groebner bases over \(\ZZ\) to find the primes modulo which a system of equations has a solution, when the system has no solutions over the rationals.

We first form a certain ideal \(I\) in \(\ZZ[x, y, z]\), and note that the Groebner basis of \(I\) over \(\QQ\) contains 1, so there are no solutions over \(\QQ\) or an algebraic closure of it (this is not surprising as there are 4 equations in 3 unknowns).

sage: P.<x,y,z> = PolynomialRing(ZZ,order='lex')
sage: I = ideal(-y^2 - 3*y + z^2 + 3, -2*y*z + z^2 + 2*z + 1,
....:           x*z + y*z + z^2, -3*x*y + 2*y*z + 6*z^2)
sage: I.change_ring(P.change_ring(QQ)).groebner_basis()
[1]

>>> from sage.all import *
>>> P = PolynomialRing(ZZ,order='lex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> I = ideal(-y**Integer(2) - Integer(3)*y + z**Integer(2) + Integer(3), -Integer(2)*y*z + z**Integer(2) + Integer(2)*z + Integer(1),
...           x*z + y*z + z**Integer(2), -Integer(3)*x*y + Integer(2)*y*z + Integer(6)*z**Integer(2))
>>> I.change_ring(P.change_ring(QQ)).groebner_basis()
[1]

However, when we compute the Groebner basis of \(I\) (defined over \(\ZZ\)), we note that there is a certain integer in the ideal which is not 1.

sage: I.groebner_basis()
[x + y + 57119*z + 4, y^2 + 3*y + 17220, y*z + ...,
 2*y + 158864, z^2 + 17223, 2*z + 41856, 164878]

>>> from sage.all import *
>>> I.groebner_basis()
[x + y + 57119*z + 4, y^2 + 3*y + 17220, y*z + ...,
 2*y + 158864, z^2 + 17223, 2*z + 41856, 164878]

Now for each prime \(p\) dividing this integer 164878, the Groebner basis of \(I\) modulo \(p\) will be non-trivial and will thus give a solution of the original system modulo \(p\).

sage: factor(164878)
2 * 7 * 11777

sage: I.change_ring(P.change_ring(GF(2))).groebner_basis()                      # needs sage.rings.finite_rings
[x + y + z, y^2 + y, y*z + y, z^2 + 1]
sage: I.change_ring(P.change_ring(GF(7))).groebner_basis()                      # needs sage.rings.finite_rings
[x - 1, y + 3, z - 2]
sage: I.change_ring(P.change_ring(GF(11777))).groebner_basis()                  # needs sage.rings.finite_rings
[x + 5633, y - 3007, z - 2626]

>>> from sage.all import *
>>> factor(Integer(164878))
2 * 7 * 11777

>>> I.change_ring(P.change_ring(GF(Integer(2)))).groebner_basis()                      # needs sage.rings.finite_rings
[x + y + z, y^2 + y, y*z + y, z^2 + 1]
>>> I.change_ring(P.change_ring(GF(Integer(7)))).groebner_basis()                      # needs sage.rings.finite_rings
[x - 1, y + 3, z - 2]
>>> I.change_ring(P.change_ring(GF(Integer(11777)))).groebner_basis()                  # needs sage.rings.finite_rings
[x + 5633, y - 3007, z - 2626]

The Groebner basis modulo any product of the prime factors is also non-trivial:

sage: I.change_ring(P.change_ring(IntegerModRing(2 * 7))).groebner_basis()
[x + ..., y^2 + 3*y, y*z + 7*y + 6, 2*y + 6, z^2 + 3, 2*z + 10]

>>> from sage.all import *
>>> I.change_ring(P.change_ring(IntegerModRing(Integer(2) * Integer(7)))).groebner_basis()
[x + ..., y^2 + 3*y, y*z + 7*y + 6, 2*y + 6, z^2 + 3, 2*z + 10]

Modulo any other prime the Groebner basis is trivial so there are no other solutions. For example:

sage: I.change_ring(P.change_ring(GF(3))).groebner_basis()                      # needs sage.rings.finite_rings
[1]

>>> from sage.all import *
>>> I.change_ring(P.change_ring(GF(Integer(3)))).groebner_basis()                      # needs sage.rings.finite_rings
[1]

Note

Sage distinguishes between lists or sequences of polynomials and ideals. Thus an ideal is not identified with a particular set of generators. For sequences of multivariate polynomials see sage.rings.polynomial.multi_polynomial_sequence.PolynomialSequence_generic.

AUTHORS:

  • William Stein: initial version

  • Kiran S. Kedlaya (2006-02-12): added Macaulay2 analogues of some Singular features

  • Martin Albrecht (2007,2008): refactoring, many Singular related functions, added plot()

  • Martin Albrecht (2009): added Groebner basis over rings functionality from Singular 3.1

  • John Perry (2012): bug fixing equality & containment of ideals

class sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal(ring, gens, coerce=True)[source]

Bases: MPolynomialIdeal_singular_repr, MPolynomialIdeal_macaulay2_repr, MPolynomialIdeal_magma_repr, Ideal_generic

Create an ideal in a multivariate polynomial ring.

INPUT:

  • ring – the ring the ideal is defined in

  • gens – list of generators for the ideal

  • coerce – whether to coerce elements to the ring ring

EXAMPLES:

sage: R.<x,y> = PolynomialRing(IntegerRing(), 2, order='lex')
sage: R.ideal([x, y])
Ideal (x, y) of Multivariate Polynomial Ring in x, y over Integer Ring
sage: R.<x0,x1> = GF(3)[]
sage: R.ideal([x0^2, x1^3])
Ideal (x0^2, x1^3) of Multivariate Polynomial Ring in x0, x1 over Finite Field of size 3

>>> from sage.all import *
>>> R = PolynomialRing(IntegerRing(), Integer(2), order='lex', names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> R.ideal([x, y])
Ideal (x, y) of Multivariate Polynomial Ring in x, y over Integer Ring
>>> R = GF(Integer(3))['x0, x1']; (x0, x1,) = R._first_ngens(2)
>>> R.ideal([x0**Integer(2), x1**Integer(3)])
Ideal (x0^2, x1^3) of Multivariate Polynomial Ring in x0, x1 over Finite Field of size 3
property basis

Shortcut to gens().

EXAMPLES:

sage: P.<x,y> = PolynomialRing(QQ,2)
sage: I = Ideal([x, y + 1])
sage: I.basis
[x, y + 1]

>>> from sage.all import *
>>> P = PolynomialRing(QQ,Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> I = Ideal([x, y + Integer(1)])
>>> I.basis
[x, y + 1]
change_ring(P)[source]

Return the ideal I in P spanned by the generators \(g_1, ..., g_n\) of self as returned by self.gens().

INPUT:

  • P – a multivariate polynomial ring

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ,3,order='lex')
sage: I = sage.rings.ideal.Cyclic(P)
sage: I
Ideal (x + y + z, x*y + x*z + y*z, x*y*z - 1) of
Multivariate Polynomial Ring in x, y, z over Rational Field

>>> from sage.all import *
>>> P = PolynomialRing(QQ,Integer(3),order='lex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> I = sage.rings.ideal.Cyclic(P)
>>> I
Ideal (x + y + z, x*y + x*z + y*z, x*y*z - 1) of
Multivariate Polynomial Ring in x, y, z over Rational Field


sage: I.groebner_basis()
[x + y + z, y^2 + y*z + z^2, z^3 - 1]

>>> from sage.all import *
>>> I.groebner_basis()
[x + y + z, y^2 + y*z + z^2, z^3 - 1]


sage: Q.<x,y,z> = P.change_ring(order='degrevlex'); Q
Multivariate Polynomial Ring in x, y, z over Rational Field
sage: Q.term_order()
Degree reverse lexicographic term order

>>> from sage.all import *
>>> Q = P.change_ring(order='degrevlex', names=('x', 'y', 'z',)); (x, y, z,) = Q._first_ngens(3); Q
Multivariate Polynomial Ring in x, y, z over Rational Field
>>> Q.term_order()
Degree reverse lexicographic term order


sage: J = I.change_ring(Q); J
Ideal (x + y + z, x*y + x*z + y*z, x*y*z - 1) of
Multivariate Polynomial Ring in x, y, z over Rational Field

>>> from sage.all import *
>>> J = I.change_ring(Q); J
Ideal (x + y + z, x*y + x*z + y*z, x*y*z - 1) of
Multivariate Polynomial Ring in x, y, z over Rational Field


sage: J.groebner_basis()
[z^3 - 1, y^2 + y*z + z^2, x + y + z]

>>> from sage.all import *
>>> J.groebner_basis()
[z^3 - 1, y^2 + y*z + z^2, x + y + z]
degree_of_semi_regularity()[source]

Return the degree of semi-regularity of this ideal under the assumption that it is semi-regular.

Let \(\{f_1, ... , f_m\} \subset K[x_1 , ... , x_n]\) be homogeneous polynomials of degrees \(d_1,... ,d_m\) respectively. This sequence is semi-regular if:

  • \(\{f_1, ... , f_m\} \neq K[x_1 , ... , x_n]\)

  • for all \(1 \leq i \leq m\) and \(g \in K[x_1,\dots,x_n]\): \(deg(g \cdot pi ) < D\) and \(g \cdot f_i \in <f_1 , \dots , f_{i-1}>\) implies that \(g \in <f_1, ..., f_{i-1}>\) where \(D\) is the degree of regularity.

This notion can be extended to affine polynomials by considering their homogeneous components of highest degree.

The degree of regularity of a semi-regular sequence \(f_1, ...,f_m\) of respective degrees \(d_1,...,d_m\) is given by the index of the first nonpositive coefficient of:

\(\sum c_k z^k = \frac{\prod (1 - z^{d_i})}{(1-z)^n}\)

EXAMPLES:

We consider a homogeneous example:

sage: n = 8
sage: K = GF(127)
sage: P = PolynomialRing(K, n, 'x')
sage: s = [K.random_element() for _ in range(n)]
sage: L = []
sage: for i in range(2 * n):
....:     f = P.random_element(degree=2, terms=binomial(n, 2))
....:     f -= f(*s)
....:     L.append(f.homogenize())
sage: I = Ideal(L)
sage: I.degree_of_semi_regularity()
4

>>> from sage.all import *
>>> n = Integer(8)
>>> K = GF(Integer(127))
>>> P = PolynomialRing(K, n, 'x')
>>> s = [K.random_element() for _ in range(n)]
>>> L = []
>>> for i in range(Integer(2) * n):
...     f = P.random_element(degree=Integer(2), terms=binomial(n, Integer(2)))
...     f -= f(*s)
...     L.append(f.homogenize())
>>> I = Ideal(L)
>>> I.degree_of_semi_regularity()
4

From this, we expect a Groebner basis computation to reach at most degree 4. For homogeneous systems this is equivalent to the largest degree in the Groebner basis:

sage: max(f.degree() for f in I.groebner_basis())
4

>>> from sage.all import *
>>> max(f.degree() for f in I.groebner_basis())
4

We increase the number of polynomials and observe a decrease the degree of regularity:

sage: for i in range(2 * n):
....:     f = P.random_element(degree=2, terms=binomial(n, 2))
....:     f -= f(*s)
....:     L.append(f.homogenize())
sage: I = Ideal(L)
sage: I.degree_of_semi_regularity()
3

sage: max(f.degree() for f in I.groebner_basis())
3

>>> from sage.all import *
>>> for i in range(Integer(2) * n):
...     f = P.random_element(degree=Integer(2), terms=binomial(n, Integer(2)))
...     f -= f(*s)
...     L.append(f.homogenize())
>>> I = Ideal(L)
>>> I.degree_of_semi_regularity()
3

>>> max(f.degree() for f in I.groebner_basis())
3

The degree of regularity approaches 2 for quadratic systems as the number of polynomials approaches \(n^2\):

sage: for i in range((n-4) * n):
....:     f = P.random_element(degree=2, terms=binomial(n, 2))
....:     f -= f(*s)
....:     L.append(f.homogenize())
sage: I = Ideal(L)
sage: I.degree_of_semi_regularity()
2

sage: max(f.degree() for f in I.groebner_basis())
2

>>> from sage.all import *
>>> for i in range((n-Integer(4)) * n):
...     f = P.random_element(degree=Integer(2), terms=binomial(n, Integer(2)))
...     f -= f(*s)
...     L.append(f.homogenize())
>>> I = Ideal(L)
>>> I.degree_of_semi_regularity()
2

>>> max(f.degree() for f in I.groebner_basis())
2

Note

It is unknown whether semi-regular sequences exist. However, it is expected that random systems are semi-regular sequences. For more details about semi-regular sequences see [BFS2004].

gens()[source]

Return a set of generators / a basis of this ideal. This is usually the set of generators provided during object creation.

EXAMPLES:

sage: P.<x,y> = PolynomialRing(QQ,2)
sage: I = Ideal([x, y + 1]); I
Ideal (x, y + 1) of Multivariate Polynomial Ring in x, y over Rational Field
sage: I.gens()
[x, y + 1]

>>> from sage.all import *
>>> P = PolynomialRing(QQ,Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> I = Ideal([x, y + Integer(1)]); I
Ideal (x, y + 1) of Multivariate Polynomial Ring in x, y over Rational Field
>>> I.gens()
[x, y + 1]
groebner_basis(algorithm='', deg_bound=None, mult_bound=None, prot=False, *args, **kwds)[source]

Return the reduced Groebner basis of this ideal.

A Groebner basis \(g_1,...,g_n\) for an ideal \(I\) is a generating set such that \(<LM(g_i)> = LM(I)\), i.e., the leading monomial ideal of \(I\) is spanned by the leading terms of \(g_1,...,g_n\). Groebner bases are the key concept in computational ideal theory in multivariate polynomial rings which allows a variety of problems to be solved.

Additionally, a reduced Groebner basis \(G\) is a unique representation for the ideal \(<G>\) with respect to the chosen monomial ordering.

INPUT:

  • algorithm – determines the algorithm to use, see below for available algorithms

  • deg_bound – only compute to degree deg_bound, that is, ignore all S-polynomials of higher degree. (default: None)

  • mult_bound – the computation is stopped if the ideal is zero-dimensional in a ring with local ordering and its multiplicity is lower than mult_bound. Singular only. (default: None)

  • prot – if set to True the computation protocol of the underlying implementation is printed. If an algorithm from the singular: or magma: family is used, prot may also be sage in which case the output is parsed and printed in a common format where the amount of information printed can be controlled via calls to set_verbose().

  • *args – additional parameters passed to the respective implementations

  • **kwds – additional keyword parameters passed to the respective implementations

ALGORITHMS:

''

autoselect (default)

'singular:groebner'

Singular’s groebner command

'singular:std'

Singular’s std command

'singular:stdhilb'

Singular’s stdhib command

'singular:stdfglm'

Singular’s stdfglm command

'singular:slimgb'

Singular’s slimgb command

'libsingular:groebner'

libSingular’s groebner command

'libsingular:std'

libSingular’s std command

'libsingular:slimgb'

libSingular’s slimgb command

'libsingular:stdhilb'

libSingular’s stdhib command

'libsingular:stdfglm'

libSingular’s stdfglm command

'toy:buchberger'

Sage’s toy/educational buchberger without Buchberger criteria

'toy:buchberger2'

Sage’s toy/educational buchberger with Buchberger criteria

'toy:d_basis'

Sage’s toy/educational algorithm for computation over PIDs

'macaulay2:gb'

Macaulay2’s gb command (if available)

'macaulay2:f4'

Macaulay2’s GroebnerBasis command with the strategy “F4” (if available)

'macaulay2:mgb'

Macaulay2’s GroebnerBasis command with the strategy “MGB” (if available)

'msolve'

optional package msolve (degrevlex order)

'magma:GroebnerBasis'

Magma’s Groebnerbasis command (if available)

'ginv:TQ', 'ginv:TQBlockHigh', 'ginv:TQBlockLow' and 'ginv:TQDegree'

One of GINV’s implementations (if available)

'giac:gbasis'

Giac’s gbasis command (if available)

If only a system is given - e.g. 'magma' - the default algorithm is chosen for that system.

Note

The Singular and libSingular versions of the respective algorithms are identical, but the former calls an external Singular process while the latter calls a C function, and thus the calling overhead is smaller. However, the libSingular interface does not support pretty printing of computation protocols.

EXAMPLES:

Consider Katsura-3 over \(\QQ\) with lexicographical term ordering. We compute the reduced Groebner basis using every available implementation and check their equality.

sage: P.<a,b,c> = PolynomialRing(QQ,3, order='lex')
sage: I = sage.rings.ideal.Katsura(P,3)  # regenerate to prevent caching
sage: I.groebner_basis()
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

>>> from sage.all import *
>>> P = PolynomialRing(QQ,Integer(3), order='lex', names=('a', 'b', 'c',)); (a, b, c,) = P._first_ngens(3)
>>> I = sage.rings.ideal.Katsura(P,Integer(3))  # regenerate to prevent caching
>>> I.groebner_basis()
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]


sage: I = sage.rings.ideal.Katsura(P,3)  # regenerate to prevent caching
sage: I.groebner_basis('libsingular:groebner')
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

>>> from sage.all import *
>>> I = sage.rings.ideal.Katsura(P,Integer(3))  # regenerate to prevent caching
>>> I.groebner_basis('libsingular:groebner')
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]


sage: I = sage.rings.ideal.Katsura(P,3)  # regenerate to prevent caching
sage: I.groebner_basis('libsingular:std')
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

>>> from sage.all import *
>>> I = sage.rings.ideal.Katsura(P,Integer(3))  # regenerate to prevent caching
>>> I.groebner_basis('libsingular:std')
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]


sage: I = sage.rings.ideal.Katsura(P,3)  # regenerate to prevent caching
sage: I.groebner_basis('libsingular:stdhilb')
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

>>> from sage.all import *
>>> I = sage.rings.ideal.Katsura(P,Integer(3))  # regenerate to prevent caching
>>> I.groebner_basis('libsingular:stdhilb')
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]


sage: I = sage.rings.ideal.Katsura(P,3)  # regenerate to prevent caching
sage: I.groebner_basis('libsingular:stdfglm')
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

>>> from sage.all import *
>>> I = sage.rings.ideal.Katsura(P,Integer(3))  # regenerate to prevent caching
>>> I.groebner_basis('libsingular:stdfglm')
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]


sage: I = sage.rings.ideal.Katsura(P,3)  # regenerate to prevent caching
sage: I.groebner_basis('libsingular:slimgb')
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

>>> from sage.all import *
>>> I = sage.rings.ideal.Katsura(P,Integer(3))  # regenerate to prevent caching
>>> I.groebner_basis('libsingular:slimgb')
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

Although Giac does support lexicographical ordering, we use degree reverse lexicographical ordering here, in order to test against Issue #21884:

sage: # needs sage.libs.giac
sage: I = sage.rings.ideal.Katsura(P,3)  # regenerate to prevent caching
sage: J = I.change_ring(P.change_ring(order='degrevlex'))
sage: gb = J.groebner_basis('giac')  # random
sage: gb
[c^3 - 79/210*c^2 + 1/30*b + 1/70*c, b^2 - 3/5*c^2 - 1/5*b + 1/5*c, b*c + 6/5*c^2 - 1/10*b - 2/5*c, a + 2*b + 2*c - 1]
sage: J.groebner_basis.set_cache(gb)
sage: ideal(J.transformed_basis()).change_ring(P).interreduced_basis()  # testing issue #21884
...[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

>>> from sage.all import *
>>> # needs sage.libs.giac
>>> I = sage.rings.ideal.Katsura(P,Integer(3))  # regenerate to prevent caching
>>> J = I.change_ring(P.change_ring(order='degrevlex'))
>>> gb = J.groebner_basis('giac')  # random
>>> gb
[c^3 - 79/210*c^2 + 1/30*b + 1/70*c, b^2 - 3/5*c^2 - 1/5*b + 1/5*c, b*c + 6/5*c^2 - 1/10*b - 2/5*c, a + 2*b + 2*c - 1]
>>> J.groebner_basis.set_cache(gb)
>>> ideal(J.transformed_basis()).change_ring(P).interreduced_basis()  # testing issue #21884
...[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

Giac’s gbasis over \(\QQ\) can benefit from a probabilistic lifting and multi threaded operations:

sage: # needs sage.libs.giac
sage: A9 = PolynomialRing(QQ, 9, 'x')
sage: I9 = sage.rings.ideal.Katsura(A9)
sage: print("possible output from giac", flush=True); I9.groebner_basis("giac", proba_epsilon=1e-7)  # long time (3s)
possible output...
Polynomial Sequence with 143 Polynomials in 9 Variables

>>> from sage.all import *
>>> # needs sage.libs.giac
>>> A9 = PolynomialRing(QQ, Integer(9), 'x')
>>> I9 = sage.rings.ideal.Katsura(A9)
>>> print("possible output from giac", flush=True); I9.groebner_basis("giac", proba_epsilon=RealNumber('1e-7'))  # long time (3s)
possible output...
Polynomial Sequence with 143 Polynomials in 9 Variables

The list of available Giac options is provided at sage.libs.giac.groebner_basis().

Note that toy:buchberger does not return the reduced Groebner basis,

sage: I = sage.rings.ideal.Katsura(P,3)  # regenerate to prevent caching
sage: gb = I.groebner_basis('toy:buchberger')
sage: gb.is_groebner()
True
sage: gb == gb.reduced()
False

>>> from sage.all import *
>>> I = sage.rings.ideal.Katsura(P,Integer(3))  # regenerate to prevent caching
>>> gb = I.groebner_basis('toy:buchberger')
>>> gb.is_groebner()
True
>>> gb == gb.reduced()
False

but that toy:buchberger2 does.

sage: I = sage.rings.ideal.Katsura(P,3)  # regenerate to prevent caching
sage: gb = I.groebner_basis('toy:buchberger2'); gb
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]
sage: gb == gb.reduced()
True

>>> from sage.all import *
>>> I = sage.rings.ideal.Katsura(P,Integer(3))  # regenerate to prevent caching
>>> gb = I.groebner_basis('toy:buchberger2'); gb
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]
>>> gb == gb.reduced()
True

Here we use Macaulay2 with three different strategies over a finite field.

sage: # optional - macaulay2
sage: R.<a,b,c> = PolynomialRing(GF(101), 3)
sage: I = sage.rings.ideal.Katsura(R,3)  # regenerate to prevent caching
sage: I.groebner_basis('macaulay2:gb')
[c^3 + 28*c^2 - 37*b + 13*c, b^2 - 41*c^2 + 20*b - 20*c,
 b*c - 19*c^2 + 10*b + 40*c, a + 2*b + 2*c - 1]
sage: I = sage.rings.ideal.Katsura(R,3)  # regenerate to prevent caching
sage: I.groebner_basis('macaulay2:f4')
[c^3 + 28*c^2 - 37*b + 13*c, b^2 - 41*c^2 + 20*b - 20*c,
 b*c - 19*c^2 + 10*b + 40*c, a + 2*b + 2*c - 1]
sage: I = sage.rings.ideal.Katsura(R,3)  # regenerate to prevent caching
sage: I.groebner_basis('macaulay2:mgb')
[c^3 + 28*c^2 - 37*b + 13*c, b^2 - 41*c^2 + 20*b - 20*c,
 b*c - 19*c^2 + 10*b + 40*c, a + 2*b + 2*c - 1]

>>> from sage.all import *
>>> # optional - macaulay2
>>> R = PolynomialRing(GF(Integer(101)), Integer(3), names=('a', 'b', 'c',)); (a, b, c,) = R._first_ngens(3)
>>> I = sage.rings.ideal.Katsura(R,Integer(3))  # regenerate to prevent caching
>>> I.groebner_basis('macaulay2:gb')
[c^3 + 28*c^2 - 37*b + 13*c, b^2 - 41*c^2 + 20*b - 20*c,
 b*c - 19*c^2 + 10*b + 40*c, a + 2*b + 2*c - 1]
>>> I = sage.rings.ideal.Katsura(R,Integer(3))  # regenerate to prevent caching
>>> I.groebner_basis('macaulay2:f4')
[c^3 + 28*c^2 - 37*b + 13*c, b^2 - 41*c^2 + 20*b - 20*c,
 b*c - 19*c^2 + 10*b + 40*c, a + 2*b + 2*c - 1]
>>> I = sage.rings.ideal.Katsura(R,Integer(3))  # regenerate to prevent caching
>>> I.groebner_basis('macaulay2:mgb')
[c^3 + 28*c^2 - 37*b + 13*c, b^2 - 41*c^2 + 20*b - 20*c,
 b*c - 19*c^2 + 10*b + 40*c, a + 2*b + 2*c - 1]

Over prime fields of small characteristic, we can also use the optional package msolve:

sage: R.<a,b,c> = PolynomialRing(GF(101), 3)
sage: I = sage.rings.ideal.Katsura(R,3)  # regenerate to prevent caching
sage: I.groebner_basis('msolve')                    # optional - msolve
[a + 2*b + 2*c - 1, b*c - 19*c^2 + 10*b + 40*c,
 b^2 - 41*c^2 + 20*b - 20*c, c^3 + 28*c^2 - 37*b + 13*c]

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(101)), Integer(3), names=('a', 'b', 'c',)); (a, b, c,) = R._first_ngens(3)
>>> I = sage.rings.ideal.Katsura(R,Integer(3))  # regenerate to prevent caching
>>> I.groebner_basis('msolve')                    # optional - msolve
[a + 2*b + 2*c - 1, b*c - 19*c^2 + 10*b + 40*c,
 b^2 - 41*c^2 + 20*b - 20*c, c^3 + 28*c^2 - 37*b + 13*c]


sage: I = sage.rings.ideal.Katsura(P,3)  # regenerate to prevent caching
sage: I.groebner_basis('magma:GroebnerBasis')       # optional - magma
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1,
 b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

>>> from sage.all import *
>>> I = sage.rings.ideal.Katsura(P,Integer(3))  # regenerate to prevent caching
>>> I.groebner_basis('magma:GroebnerBasis')       # optional - magma
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1,
 b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

Singular and libSingular can compute Groebner basis with degree restrictions.

sage: R.<x,y> = QQ[]
sage: I = R*[x^3 + y^2, x^2*y + 1]
sage: I.groebner_basis(algorithm='singular')
[x^3 + y^2, x^2*y + 1, y^3 - x]
sage: I.groebner_basis(algorithm='singular', deg_bound=2)
[x^3 + y^2, x^2*y + 1]
sage: I.groebner_basis()
[x^3 + y^2, x^2*y + 1, y^3 - x]
sage: I.groebner_basis(deg_bound=2)
[x^3 + y^2, x^2*y + 1]

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> I = R*[x**Integer(3) + y**Integer(2), x**Integer(2)*y + Integer(1)]
>>> I.groebner_basis(algorithm='singular')
[x^3 + y^2, x^2*y + 1, y^3 - x]
>>> I.groebner_basis(algorithm='singular', deg_bound=Integer(2))
[x^3 + y^2, x^2*y + 1]
>>> I.groebner_basis()
[x^3 + y^2, x^2*y + 1, y^3 - x]
>>> I.groebner_basis(deg_bound=Integer(2))
[x^3 + y^2, x^2*y + 1]

A protocol is printed, if the verbosity level is at least 2, or if the argument prot is provided. Historically, the protocol did not appear during doctests, so, we skip the examples with protocol output.

sage: from sage.misc.verbose import set_verbose
sage: set_verbose(2)
sage: I = R*[x^3+y^2,x^2*y+1]
sage: I.groebner_basis()  # not tested
std in (QQ),(x,y),(dp(2),C)
[...:2]3ss4s6
(S:2)--
product criterion:1 chain criterion:0
[x^3 + y^2, x^2*y + 1, y^3 - x]
sage: I.groebner_basis(prot=False)
std in (QQ),(x,y),(dp(2),C)
[...:2]3ss4s6
(S:2)--
product criterion:1 chain criterion:0
[x^3 + y^2, x^2*y + 1, y^3 - x]
sage: set_verbose(0)
sage: I.groebner_basis(prot=True)  # not tested
std in (QQ),(x,y),(dp(2),C)
[...:2]3ss4s6
(S:2)--
product criterion:1 chain criterion:0
[x^3 + y^2, x^2*y + 1, y^3 - x]

>>> from sage.all import *
>>> from sage.misc.verbose import set_verbose
>>> set_verbose(Integer(2))
>>> I = R*[x**Integer(3)+y**Integer(2),x**Integer(2)*y+Integer(1)]
>>> I.groebner_basis()  # not tested
std in (QQ),(x,y),(dp(2),C)
[...:2]3ss4s6
(S:2)--
product criterion:1 chain criterion:0
[x^3 + y^2, x^2*y + 1, y^3 - x]
>>> I.groebner_basis(prot=False)
std in (QQ),(x,y),(dp(2),C)
[...:2]3ss4s6
(S:2)--
product criterion:1 chain criterion:0
[x^3 + y^2, x^2*y + 1, y^3 - x]
>>> set_verbose(Integer(0))
>>> I.groebner_basis(prot=True)  # not tested
std in (QQ),(x,y),(dp(2),C)
[...:2]3ss4s6
(S:2)--
product criterion:1 chain criterion:0
[x^3 + y^2, x^2*y + 1, y^3 - x]

The list of available options is provided at LibSingularOptions.

Note that Groebner bases over \(\ZZ\) can also be computed.

sage: P.<a,b,c> = PolynomialRing(ZZ,3)
sage: I = P * (a + 2*b + 2*c - 1, a^2 - a + 2*b^2 + 2*c^2, 2*a*b + 2*b*c - b)
sage: I.groebner_basis()
[b^3 + b*c^2 + 12*c^3 + b^2 + b*c - 4*c^2,
 2*b*c^2 - 6*c^3 - b^2 - b*c + 2*c^2,
 42*c^3 + b^2 + 2*b*c - 14*c^2 + b,
 2*b^2 + 6*b*c + 6*c^2 - b - 2*c,
 10*b*c + 12*c^2 - b - 4*c,
 a + 2*b + 2*c - 1]

>>> from sage.all import *
>>> P = PolynomialRing(ZZ,Integer(3), names=('a', 'b', 'c',)); (a, b, c,) = P._first_ngens(3)
>>> I = P * (a + Integer(2)*b + Integer(2)*c - Integer(1), a**Integer(2) - a + Integer(2)*b**Integer(2) + Integer(2)*c**Integer(2), Integer(2)*a*b + Integer(2)*b*c - b)
>>> I.groebner_basis()
[b^3 + b*c^2 + 12*c^3 + b^2 + b*c - 4*c^2,
 2*b*c^2 - 6*c^3 - b^2 - b*c + 2*c^2,
 42*c^3 + b^2 + 2*b*c - 14*c^2 + b,
 2*b^2 + 6*b*c + 6*c^2 - b - 2*c,
 10*b*c + 12*c^2 - b - 4*c,
 a + 2*b + 2*c - 1]


sage: I.groebner_basis('macaulay2')                 # optional - macaulay2
[b^3 + b*c^2 + 12*c^3 + b^2 + b*c - 4*c^2,
 2*b*c^2 - 6*c^3 + b^2 + 5*b*c + 8*c^2 - b - 2*c,
 42*c^3 + b^2 + 2*b*c - 14*c^2 + b,
 2*b^2 - 4*b*c - 6*c^2 + 2*c, 10*b*c + 12*c^2 - b - 4*c,
 a + 2*b + 2*c - 1]

>>> from sage.all import *
>>> I.groebner_basis('macaulay2')                 # optional - macaulay2
[b^3 + b*c^2 + 12*c^3 + b^2 + b*c - 4*c^2,
 2*b*c^2 - 6*c^3 + b^2 + 5*b*c + 8*c^2 - b - 2*c,
 42*c^3 + b^2 + 2*b*c - 14*c^2 + b,
 2*b^2 - 4*b*c - 6*c^2 + 2*c, 10*b*c + 12*c^2 - b - 4*c,
 a + 2*b + 2*c - 1]

Groebner bases over \(\ZZ/n\ZZ\) are also supported:

sage: P.<a,b,c> = PolynomialRing(Zmod(1000), 3)
sage: I = P * (a + 2*b + 2*c - 1, a^2 - a + 2*b^2 + 2*c^2, 2*a*b + 2*b*c - b)
sage: I.groebner_basis()
[b*c^2 + 732*b*c + 808*b,
 2*c^3 + 884*b*c + 666*c^2 + 320*b,
 b^2 + 438*b*c + 281*b,
 5*b*c + 156*c^2 + 112*b + 948*c,
 50*c^2 + 600*b + 650*c,
 a + 2*b + 2*c + 999,
 125*b]

>>> from sage.all import *
>>> P = PolynomialRing(Zmod(Integer(1000)), Integer(3), names=('a', 'b', 'c',)); (a, b, c,) = P._first_ngens(3)
>>> I = P * (a + Integer(2)*b + Integer(2)*c - Integer(1), a**Integer(2) - a + Integer(2)*b**Integer(2) + Integer(2)*c**Integer(2), Integer(2)*a*b + Integer(2)*b*c - b)
>>> I.groebner_basis()
[b*c^2 + 732*b*c + 808*b,
 2*c^3 + 884*b*c + 666*c^2 + 320*b,
 b^2 + 438*b*c + 281*b,
 5*b*c + 156*c^2 + 112*b + 948*c,
 50*c^2 + 600*b + 650*c,
 a + 2*b + 2*c + 999,
 125*b]


sage: R.<x,y,z> = PolynomialRing(Zmod(2233497349584))
sage: I = R.ideal([z*(x-3*y), 3^2*x^2-y*z, z^2+y^2])
sage: I.groebner_basis()
[2*z^4, y*z^2 + 81*z^3, 248166372176*z^3, 9*x^2 - y*z, y^2 + z^2, x*z +
2233497349581*y*z, 248166372176*y*z]

>>> from sage.all import *
>>> R = PolynomialRing(Zmod(Integer(2233497349584)), names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> I = R.ideal([z*(x-Integer(3)*y), Integer(3)**Integer(2)*x**Integer(2)-y*z, z**Integer(2)+y**Integer(2)])
>>> I.groebner_basis()
[2*z^4, y*z^2 + 81*z^3, 248166372176*z^3, 9*x^2 - y*z, y^2 + z^2, x*z +
2233497349581*y*z, 248166372176*y*z]

Sage also supports local orderings:

sage: P.<x,y,z> = PolynomialRing(QQ, 3, order='negdegrevlex')
sage: I = P * (  x*y*z + z^5, 2*x^2 + y^3 + z^7, 3*z^5 + y^5 )
sage: I.groebner_basis()
[x^2 + 1/2*y^3, x*y*z + z^5, y^5 + 3*z^5, y^4*z - 2*x*z^5, z^6]

>>> from sage.all import *
>>> P = PolynomialRing(QQ, Integer(3), order='negdegrevlex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> I = P * (  x*y*z + z**Integer(5), Integer(2)*x**Integer(2) + y**Integer(3) + z**Integer(7), Integer(3)*z**Integer(5) + y**Integer(5) )
>>> I.groebner_basis()
[x^2 + 1/2*y^3, x*y*z + z^5, y^5 + 3*z^5, y^4*z - 2*x*z^5, z^6]

We can represent every element in the ideal as a combination of the generators using the lift() method:

sage: P.<x,y,z> = PolynomialRing(QQ, 3)
sage: I = P * ( x*y*z + z^5, 2*x^2 + y^3 + z^7, 3*z^5 + y^5 )
sage: J = Ideal(I.groebner_basis())
sage: f = sum(P.random_element(terms=2)*f for f in I.gens())
sage: f                        # random
1/2*y^2*z^7 - 1/4*y*z^8 + 2*x*z^5 + 95*z^6 + 1/2*y^5 - 1/4*y^4*z + x^2*y^2 + 3/2*x^2*y*z + 95*x*y*z^2
sage: f.lift(I.gens())         # random
[2*x + 95*z, 1/2*y^2 - 1/4*y*z, 0]
sage: l = f.lift(J.gens()); l  # random
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1/2*y^2 + 1/4*y*z, 1/2*y^2*z^2 - 1/4*y*z^3 + 2*x + 95*z]
sage: sum(map(mul, zip(l,J.gens()))) == f
True

>>> from sage.all import *
>>> P = PolynomialRing(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> I = P * ( x*y*z + z**Integer(5), Integer(2)*x**Integer(2) + y**Integer(3) + z**Integer(7), Integer(3)*z**Integer(5) + y**Integer(5) )
>>> J = Ideal(I.groebner_basis())
>>> f = sum(P.random_element(terms=Integer(2))*f for f in I.gens())
>>> f                        # random
1/2*y^2*z^7 - 1/4*y*z^8 + 2*x*z^5 + 95*z^6 + 1/2*y^5 - 1/4*y^4*z + x^2*y^2 + 3/2*x^2*y*z + 95*x*y*z^2
>>> f.lift(I.gens())         # random
[2*x + 95*z, 1/2*y^2 - 1/4*y*z, 0]
>>> l = f.lift(J.gens()); l  # random
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1/2*y^2 + 1/4*y*z, 1/2*y^2*z^2 - 1/4*y*z^3 + 2*x + 95*z]
>>> sum(map(mul, zip(l,J.gens()))) == f
True

Groebner bases over fraction fields of polynomial rings are also supported:

sage: P.<t> = QQ[]
sage: F = Frac(P)
sage: R.<X,Y,Z> = F[]
sage: I = Ideal([f + P.random_element() for f in sage.rings.ideal.Katsura(R).gens()])
sage: I.groebner_basis().ideal() == I
True

>>> from sage.all import *
>>> P = QQ['t']; (t,) = P._first_ngens(1)
>>> F = Frac(P)
>>> R = F['X, Y, Z']; (X, Y, Z,) = R._first_ngens(3)
>>> I = Ideal([f + P.random_element() for f in sage.rings.ideal.Katsura(R).gens()])
>>> I.groebner_basis().ideal() == I
True

In cases where a characteristic cannot be determined, we use a toy implementation of Buchberger’s algorithm (see Issue #6581):

sage: R.<a,b> = QQ[]; I = R.ideal(a^2+b^2-1)
sage: Q = QuotientRing(R,I); K = Frac(Q)
sage: R2.<x,y> = K[]; J = R2.ideal([(a^2+b^2)*x + y, x+y])
sage: J.groebner_basis()
verbose 0 (...: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
[x + y]

>>> from sage.all import *
>>> R = QQ['a, b']; (a, b,) = R._first_ngens(2); I = R.ideal(a**Integer(2)+b**Integer(2)-Integer(1))
>>> Q = QuotientRing(R,I); K = Frac(Q)
>>> R2 = K['x, y']; (x, y,) = R2._first_ngens(2); J = R2.ideal([(a**Integer(2)+b**Integer(2))*x + y, x+y])
>>> J.groebner_basis()
verbose 0 (...: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
[x + y]

ALGORITHM:

Uses Singular, one of the other systems listed above (if available), or a toy implementation.

groebner_cover()[source]

Compute the Gröbner cover of the ideal, over a field with parameters.

The Groebner cover is a partition of the space of parameters, such that the Groebner basis in each part is given by the same expression.

EXAMPLES:

sage: F = PolynomialRing(QQ,'a').fraction_field()
sage: F.inject_variables()
Defining a
sage: R.<x,y,z> = F[]
sage: I = R.ideal([-x+3*y+z-5,2*x+a*z+4,4*x-3*z-1/a])
sage: I.groebner_cover()
{Quasi-affine subscheme X - Y of Affine Space of dimension 1 over Rational Field,
   where X is defined by:
     0
   and Y is defined by:
     2*a^2 + 3*a: [(2*a^2 + 3*a)*z + (8*a + 1),
                   (12*a^2 + 18*a)*y + (-20*a^2 - 35*a - 2), (4*a + 6)*x + 11],
 Quasi-affine subscheme X - Y of Affine Space of dimension 1 over Rational Field,
   where X is defined by:
     ...
   and Y is defined by:
     1: [1],
 Quasi-affine subscheme X - Y of Affine Space of dimension 1 over Rational Field,
   where X is defined by:
     ...
   and Y is defined by:
     1: [1]}

>>> from sage.all import *
>>> F = PolynomialRing(QQ,'a').fraction_field()
>>> F.inject_variables()
Defining a
>>> R = F['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> I = R.ideal([-x+Integer(3)*y+z-Integer(5),Integer(2)*x+a*z+Integer(4),Integer(4)*x-Integer(3)*z-Integer(1)/a])
>>> I.groebner_cover()
{Quasi-affine subscheme X - Y of Affine Space of dimension 1 over Rational Field,
   where X is defined by:
     0
   and Y is defined by:
     2*a^2 + 3*a: [(2*a^2 + 3*a)*z + (8*a + 1),
                   (12*a^2 + 18*a)*y + (-20*a^2 - 35*a - 2), (4*a + 6)*x + 11],
 Quasi-affine subscheme X - Y of Affine Space of dimension 1 over Rational Field,
   where X is defined by:
     ...
   and Y is defined by:
     1: [1],
 Quasi-affine subscheme X - Y of Affine Space of dimension 1 over Rational Field,
   where X is defined by:
     ...
   and Y is defined by:
     1: [1]}
groebner_fan(is_groebner_basis=False, symmetry=None, verbose=False)[source]

Return the Groebner fan of this ideal.

The base ring must be \(\QQ\) or a finite field \(\GF{p}\) of with \(p \leq 32749\).

EXAMPLES:

sage: P.<x,y> = PolynomialRing(QQ)
sage: i = ideal(x^2 - y^2 + 1)
sage: g = i.groebner_fan()
sage: g.reduced_groebner_bases()
[[x^2 - y^2 + 1], [-x^2 + y^2 - 1]]

>>> from sage.all import *
>>> P = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> i = ideal(x**Integer(2) - y**Integer(2) + Integer(1))
>>> g = i.groebner_fan()
>>> g.reduced_groebner_bases()
[[x^2 - y^2 + 1], [-x^2 + y^2 - 1]]

INPUT:

  • is_groebner_basis – boolean (default: False); if True, then I.gens() must be a Groebner basis with respect to the standard degree lexicographic term order

  • symmetry – (default: None) if not None, describes symmetries of the ideal

  • verbose – (default: False) if True, printout useful info during computations

homogenize(var='h')[source]

Return homogeneous ideal spanned by the homogeneous polynomials generated by homogenizing the generators of this ideal.

INPUT:

  • h – variable name or variable in cover ring (default: 'h')

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(GF(2))
sage: I = Ideal([x^2*y + z + 1, x + y^2 + 1]); I
Ideal (x^2*y + z + 1, y^2 + x + 1) of Multivariate
Polynomial Ring in x, y, z over Finite Field of size 2

>>> from sage.all import *
>>> P = PolynomialRing(GF(Integer(2)), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> I = Ideal([x**Integer(2)*y + z + Integer(1), x + y**Integer(2) + Integer(1)]); I
Ideal (x^2*y + z + 1, y^2 + x + 1) of Multivariate
Polynomial Ring in x, y, z over Finite Field of size 2


sage: I.homogenize()
Ideal (x^2*y + z*h^2 + h^3, y^2 + x*h + h^2) of
Multivariate Polynomial Ring in x, y, z, h over Finite
Field of size 2

>>> from sage.all import *
>>> I.homogenize()
Ideal (x^2*y + z*h^2 + h^3, y^2 + x*h + h^2) of
Multivariate Polynomial Ring in x, y, z, h over Finite
Field of size 2


sage: I.homogenize(y)
Ideal (x^2*y + y^3 + y^2*z, x*y) of Multivariate
Polynomial Ring in x, y, z over Finite Field of size 2

>>> from sage.all import *
>>> I.homogenize(y)
Ideal (x^2*y + y^3 + y^2*z, x*y) of Multivariate
Polynomial Ring in x, y, z over Finite Field of size 2


sage: I = Ideal([x^2*y + z^3 + y^2*x, x + y^2 + 1])
sage: I.homogenize()
Ideal (x^2*y + x*y^2 + z^3, y^2 + x*h + h^2) of
Multivariate Polynomial Ring in x, y, z, h over Finite
Field of size 2

>>> from sage.all import *
>>> I = Ideal([x**Integer(2)*y + z**Integer(3) + y**Integer(2)*x, x + y**Integer(2) + Integer(1)])
>>> I.homogenize()
Ideal (x^2*y + x*y^2 + z^3, y^2 + x*h + h^2) of
Multivariate Polynomial Ring in x, y, z, h over Finite
Field of size 2
is_homogeneous()[source]

Return True if this ideal is spanned by homogeneous polynomials, i.e., if it is a homogeneous ideal.

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ,3)
sage: I = sage.rings.ideal.Katsura(P)
sage: I
Ideal (x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y +
2*y*z - y) of Multivariate Polynomial Ring in x, y, z over
Rational Field

>>> from sage.all import *
>>> P = PolynomialRing(QQ,Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> I = sage.rings.ideal.Katsura(P)
>>> I
Ideal (x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y +
2*y*z - y) of Multivariate Polynomial Ring in x, y, z over
Rational Field


sage: I.is_homogeneous()
False

>>> from sage.all import *
>>> I.is_homogeneous()
False


sage: J = I.homogenize()
sage: J
Ideal (x + 2*y + 2*z - h, x^2 + 2*y^2 + 2*z^2 - x*h, 2*x*y
+ 2*y*z - y*h) of Multivariate Polynomial Ring in x, y, z,
h over Rational Field

>>> from sage.all import *
>>> J = I.homogenize()
>>> J
Ideal (x + 2*y + 2*z - h, x^2 + 2*y^2 + 2*z^2 - x*h, 2*x*y
+ 2*y*z - y*h) of Multivariate Polynomial Ring in x, y, z,
h over Rational Field


sage: J.is_homogeneous()
True

>>> from sage.all import *
>>> J.is_homogeneous()
True
plot(*args, **kwds)[source]

Plot the real zero locus of this principal ideal.

INPUT:

  • self – a principal ideal in 2 variables

  • algorithm – set this to ‘surf’ if you want ‘surf’ to plot the ideal (default: None)

  • *args – (optional) tuples (variable, minimum, maximum) for plotting dimensions

  • **kwds – optional keyword arguments passed on to implicit_plot

EXAMPLES:

Implicit plotting in 2-d:

sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: I = R.ideal([y^3 - x^2])
sage: I.plot()                         # cusp                               # needs sage.plot
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(2), names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> I = R.ideal([y**Integer(3) - x**Integer(2)])
>>> I.plot()                         # cusp                               # needs sage.plot
Graphics object consisting of 1 graphics primitive


sage: I = R.ideal([y^2 - x^2 - 1])
sage: I.plot((x,-3, 3), (y, -2, 2))    # hyperbola                          # needs sage.plot
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> I = R.ideal([y**Integer(2) - x**Integer(2) - Integer(1)])
>>> I.plot((x,-Integer(3), Integer(3)), (y, -Integer(2), Integer(2)))    # hyperbola                          # needs sage.plot
Graphics object consisting of 1 graphics primitive


sage: I = R.ideal([y^2 + x^2*(1/4) - 1])
sage: I.plot()                         # ellipse                            # needs sage.plot
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> I = R.ideal([y**Integer(2) + x**Integer(2)*(Integer(1)/Integer(4)) - Integer(1)])
>>> I.plot()                         # ellipse                            # needs sage.plot
Graphics object consisting of 1 graphics primitive


sage: I = R.ideal([y^2-(x^2-1)*(x-2)])
sage: I.plot()                         # elliptic curve                     # needs sage.plot
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> I = R.ideal([y**Integer(2)-(x**Integer(2)-Integer(1))*(x-Integer(2))])
>>> I.plot()                         # elliptic curve                     # needs sage.plot
Graphics object consisting of 1 graphics primitive


sage: f = ((x+3)^3 + 2*(x+3)^2 - y^2)*(x^3 - y^2)*((x-3)^3-2*(x-3)^2-y^2)
sage: I = R.ideal(f)
sage: I.plot()                         # the Singular logo                  # needs sage.plot
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> f = ((x+Integer(3))**Integer(3) + Integer(2)*(x+Integer(3))**Integer(2) - y**Integer(2))*(x**Integer(3) - y**Integer(2))*((x-Integer(3))**Integer(3)-Integer(2)*(x-Integer(3))**Integer(2)-y**Integer(2))
>>> I = R.ideal(f)
>>> I.plot()                         # the Singular logo                  # needs sage.plot
Graphics object consisting of 1 graphics primitive


sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: I = R.ideal([x - 1])
sage: I.plot((y, -2, 2))               # vertical line                      # needs sage.plot
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(2), names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> I = R.ideal([x - Integer(1)])
>>> I.plot((y, -Integer(2), Integer(2)))               # vertical line                      # needs sage.plot
Graphics object consisting of 1 graphics primitive


sage: I = R.ideal([-x^2*y + 1])
sage: I.plot()                         # blow up                            # needs sage.plot
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> I = R.ideal([-x**Integer(2)*y + Integer(1)])
>>> I.plot()                         # blow up                            # needs sage.plot
Graphics object consisting of 1 graphics primitive
random_element(degree, compute_gb=False, *args, **kwds)[source]

Return a random element in this ideal as \(r = \sum h_i·f_i\).

INPUT:

  • compute_gb – if True then a Gröbner basis is computed first and \(f_i\) are the elements in the Gröbner basis. Otherwise whatever basis is returned by self.gens() is used.

  • *args and **kwds are passed to R.random_element() with R = self.ring().

EXAMPLES:

We compute a uniformly random element up to the provided degree.

sage: P.<x,y,z> = GF(127)[]
sage: I = sage.rings.ideal.Katsura(P)
sage: f = I.random_element(degree=4, compute_gb=True, terms=infinity)
sage: f.degree() <= 4
True
sage: len(list(f)) <= 35
True

>>> from sage.all import *
>>> P = GF(Integer(127))['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> I = sage.rings.ideal.Katsura(P)
>>> f = I.random_element(degree=Integer(4), compute_gb=True, terms=infinity)
>>> f.degree() <= Integer(4)
True
>>> len(list(f)) <= Integer(35)
True

Note that sampling uniformly at random from the ideal at some large enough degree is equivalent to computing a Gröbner basis. We give an example showing how to compute a Gröbner basis if we can sample uniformly at random from an ideal:

sage: n = 3; d = 4
sage: P = PolynomialRing(GF(127), n, 'x')
sage: I = sage.rings.ideal.Cyclic(P)

>>> from sage.all import *
>>> n = Integer(3); d = Integer(4)
>>> P = PolynomialRing(GF(Integer(127)), n, 'x')
>>> I = sage.rings.ideal.Cyclic(P)

  1. We sample \(n^d\) uniformly random elements in the ideal:

    sage: F = Sequence(I.random_element(degree=d, compute_gb=True,
....:                               terms=infinity)
....:              for _ in range(n^d))

    >>> from sage.all import *
>>> F = Sequence(I.random_element(degree=d, compute_gb=True,
...                               terms=infinity)
...              for _ in range(n**d))

  2. We linearize and compute the echelon form:

    sage: A, v = F.coefficients_monomials()
sage: A.echelonize()

    >>> from sage.all import *
>>> A, v = F.coefficients_monomials()
>>> A.echelonize()

  3. The result is the desired Gröbner basis:

    sage: G = Sequence((A * v).list())
sage: G.is_groebner()
True
sage: Ideal(G) == I
True

    >>> from sage.all import *
>>> G = Sequence((A * v).list())
>>> G.is_groebner()
True
>>> Ideal(G) == I
True

We return some element in the ideal with no guarantee on the distribution:

sage: # needs sage.rings.finite_rings
sage: P = PolynomialRing(GF(127), 10, 'x')
sage: I = sage.rings.ideal.Katsura(P)
sage: f = I.random_element(degree=3)
sage: f  # random
-25*x0^2*x1 + 14*x1^3 + 57*x0*x1*x2 + ... + 19*x7*x9 + 40*x8*x9 + 49*x1
sage: f.degree()
3

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> P = PolynomialRing(GF(Integer(127)), Integer(10), 'x')
>>> I = sage.rings.ideal.Katsura(P)
>>> f = I.random_element(degree=Integer(3))
>>> f  # random
-25*x0^2*x1 + 14*x1^3 + 57*x0*x1*x2 + ... + 19*x7*x9 + 40*x8*x9 + 49*x1
>>> f.degree()
3

We show that the default method does not sample uniformly at random from the ideal:

sage: # needs sage.rings.finite_rings
sage: P.<x,y,z> = GF(127)[]
sage: G = Sequence([x + 7, y - 2, z + 110])
sage: I = Ideal([sum(P.random_element() * g for g in G)
....:            for _ in range(4)])
sage: all(I.random_element(degree=1) == 0 for _ in range(100))
True

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> P = GF(Integer(127))['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> G = Sequence([x + Integer(7), y - Integer(2), z + Integer(110)])
>>> I = Ideal([sum(P.random_element() * g for g in G)
...            for _ in range(Integer(4))])
>>> all(I.random_element(degree=Integer(1)) == Integer(0) for _ in range(Integer(100)))
True

If degree equals the degree of the generators, a random linear combination of the generators is returned:

sage: P.<x,y> = QQ[]
sage: I = P.ideal([x^2,y^2])
sage: set_random_seed(5)
sage: I.random_element(degree=2)
-2*x^2 + 2*y^2

>>> from sage.all import *
>>> P = QQ['x, y']; (x, y,) = P._first_ngens(2)
>>> I = P.ideal([x**Integer(2),y**Integer(2)])
>>> set_random_seed(Integer(5))
>>> I.random_element(degree=Integer(2))
-2*x^2 + 2*y^2
reduce(f)[source]

Reduce an element modulo the reduced Groebner basis for this ideal. This returns 0 if and only if the element is in this ideal. In any case, this reduction is unique up to monomial orders.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: I = (x^3 + y, y) * R
sage: I.reduce(y)
0
sage: I.reduce(x^3)
0
sage: I.reduce(x - y)
x

sage: I = (y^2 - (x^3 + x)) * R
sage: I.reduce(x^3)
y^2 - x
sage: I.reduce(x^6)
y^4 - 2*x*y^2 + x^2
sage: (y^2 - x)^2
y^4 - 2*x*y^2 + x^2

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(2), names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> I = (x**Integer(3) + y, y) * R
>>> I.reduce(y)
0
>>> I.reduce(x**Integer(3))
0
>>> I.reduce(x - y)
x

>>> I = (y**Integer(2) - (x**Integer(3) + x)) * R
>>> I.reduce(x**Integer(3))
y^2 - x
>>> I.reduce(x**Integer(6))
y^4 - 2*x*y^2 + x^2
>>> (y**Integer(2) - x)**Integer(2)
y^4 - 2*x*y^2 + x^2

Note

Requires computation of a Groebner basis, which can be a very expensive operation.

subs(in_dict=None, **kwds)[source]

Substitute variables.

This method substitutes some variables in the polynomials that generate the ideal with given values. Variables that are not specified in the input remain unchanged.

INPUT:

  • in_dict – (optional) dictionary of inputs

  • **kwds – named parameters

OUTPUT:

A new ideal with modified generators. If possible, in the same polynomial ring. Raises a TypeError if no common polynomial ring of the substituted generators can be found.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(ZZ, 2, 'xy')
sage: I = R.ideal(x^5 + y^5, x^2 + y + x^2*y^2 + 5); I
Ideal (x^5 + y^5, x^2*y^2 + x^2 + y + 5)
 of Multivariate Polynomial Ring in x, y over Integer Ring
sage: I.subs(x=y)
Ideal (2*y^5, y^4 + y^2 + y + 5)
 of Multivariate Polynomial Ring in x, y over Integer Ring
sage: I.subs({x: y})    # same substitution but with dictionary
Ideal (2*y^5, y^4 + y^2 + y + 5)
 of Multivariate Polynomial Ring in x, y over Integer Ring

>>> from sage.all import *
>>> R = PolynomialRing(ZZ, Integer(2), 'xy', names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> I = R.ideal(x**Integer(5) + y**Integer(5), x**Integer(2) + y + x**Integer(2)*y**Integer(2) + Integer(5)); I
Ideal (x^5 + y^5, x^2*y^2 + x^2 + y + 5)
 of Multivariate Polynomial Ring in x, y over Integer Ring
>>> I.subs(x=y)
Ideal (2*y^5, y^4 + y^2 + y + 5)
 of Multivariate Polynomial Ring in x, y over Integer Ring
>>> I.subs({x: y})    # same substitution but with dictionary
Ideal (2*y^5, y^4 + y^2 + y + 5)
 of Multivariate Polynomial Ring in x, y over Integer Ring

The new ideal can be in a different ring:

sage: R.<a,b> = PolynomialRing(QQ, 2)
sage: S.<x,y> = PolynomialRing(QQ, 2)
sage: I = R.ideal(a^2 + b^2 + a - b + 2); I
Ideal (a^2 + b^2 + a - b + 2)
 of Multivariate Polynomial Ring in a, b over Rational Field
sage: I.subs(a=x, b=y)
Ideal (x^2 + y^2 + x - y + 2)
 of Multivariate Polynomial Ring in x, y over Rational Field

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(2), names=('a', 'b',)); (a, b,) = R._first_ngens(2)
>>> S = PolynomialRing(QQ, Integer(2), names=('x', 'y',)); (x, y,) = S._first_ngens(2)
>>> I = R.ideal(a**Integer(2) + b**Integer(2) + a - b + Integer(2)); I
Ideal (a^2 + b^2 + a - b + 2)
 of Multivariate Polynomial Ring in a, b over Rational Field
>>> I.subs(a=x, b=y)
Ideal (x^2 + y^2 + x - y + 2)
 of Multivariate Polynomial Ring in x, y over Rational Field

The resulting ring need not be a multivariate polynomial ring:

sage: T.<t> = PolynomialRing(QQ)
sage: I.subs(a=t, b=t)
Principal ideal (t^2 + 1) of Univariate Polynomial Ring in t over Rational Field
sage: var("z")                                                              # needs sage.symbolic
z
sage: I.subs(a=z, b=z)                                                      # needs sage.symbolic
Principal ideal (2*z^2 + 2) of Symbolic Ring

>>> from sage.all import *
>>> T = PolynomialRing(QQ, names=('t',)); (t,) = T._first_ngens(1)
>>> I.subs(a=t, b=t)
Principal ideal (t^2 + 1) of Univariate Polynomial Ring in t over Rational Field
>>> var("z")                                                              # needs sage.symbolic
z
>>> I.subs(a=z, b=z)                                                      # needs sage.symbolic
Principal ideal (2*z^2 + 2) of Symbolic Ring

Variables that are not substituted remain unchanged:

sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: I = R.ideal(x^2 + y^2 + x - y + 2); I
Ideal (x^2 + y^2 + x - y + 2)
 of Multivariate Polynomial Ring in x, y over Rational Field
sage: I.subs(x=1)
Ideal (y^2 - y + 4) of Multivariate Polynomial Ring in x, y over Rational Field

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(2), names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> I = R.ideal(x**Integer(2) + y**Integer(2) + x - y + Integer(2)); I
Ideal (x^2 + y^2 + x - y + 2)
 of Multivariate Polynomial Ring in x, y over Rational Field
>>> I.subs(x=Integer(1))
Ideal (y^2 - y + 4) of Multivariate Polynomial Ring in x, y over Rational Field
weil_restriction()[source]

Compute the Weil restriction of this ideal over some extension field. If the field is a finite field, then this computes the Weil restriction to the prime subfield.

A Weil restriction of scalars - denoted \(Res_{L/k}\) - is a functor which, for any finite extension of fields \(L/k\) and any algebraic variety \(X\) over \(L\), produces another corresponding variety \(Res_{L/k}(X)\), defined over \(k\). It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields.

This function does not compute this Weil restriction directly but computes on generating sets of polynomial ideals:

Let \(d\) be the degree of the field extension \(L/k\), let \(a\) a generator of \(L/k\) and \(p\) the minimal polynomial of \(L/k\). Denote this ideal by \(I\).

Specifically, this function first maps each variable \(x\) to its representation over \(k\): \(\sum_{i=0}^{d-1} a^i x_i\). Then each generator of \(I\) is evaluated over these representations and reduced modulo the minimal polynomial \(p\). The result is interpreted as a univariate polynomial in \(a\) and its coefficients are the new generators of the returned ideal.

If the input and the output ideals are radical, this is equivalent to the statement about algebraic varieties above.

OUTPUT: MPolynomialIdeal

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: k.<a> = GF(2^2)
sage: P.<x,y> = PolynomialRing(k, 2)
sage: I = Ideal([x*y + 1, a*x + 1])
sage: I.variety()
[{y: a, x: a + 1}]
sage: J = I.weil_restriction()
sage: J
Ideal (x0*y0 + x1*y1 + 1, x1*y0 + x0*y1 + x1*y1, x1 + 1, x0 + x1) of
 Multivariate Polynomial Ring in x0, x1, y0, y1 over Finite Field of size 2
sage: J += sage.rings.ideal.FieldIdeal(J.ring())  # ensure radical ideal
sage: J.variety()
[{y1: 1, y0: 0, x1: 1, x0: 1}]
sage: J.weil_restriction()  # returns J
Ideal (x0*y0 + x1*y1 + 1, x1*y0 + x0*y1 + x1*y1, x1 + 1, x0 + x1,
 x0^2 + x0, x1^2 + x1, y0^2 + y0, y1^2 + y1) of Multivariate
 Polynomial Ring in x0, x1, y0, y1 over Finite Field of size 2

sage: # needs sage.rings.finite_rings
sage: k.<a> = GF(3^5)
sage: P.<x,y,z> = PolynomialRing(k)
sage: I = sage.rings.ideal.Katsura(P)
sage: I.dimension()
0
sage: I.variety()
 [{z: 0, y: 0, x: 1}]
sage: J = I.weil_restriction(); J
Ideal (x0 - y0 - z0 - 1,
       x1 - y1 - z1, x2 - y2 - z2, x3 - y3 - z3, x4 - y4 - z4,
       x0^2 + x2*x3 + x1*x4 - y0^2 - y2*y3 - y1*y4 - z0^2 - z2*z3 - z1*z4 - x0,
       -x0*x1 - x2*x3 - x3^2 - x1*x4 + x2*x4 + y0*y1 + y2*y3
            + y3^2 + y1*y4 - y2*y4 + z0*z1 + z2*z3 + z3^2 + z1*z4 - z2*z4 - x1,
       x1^2 - x0*x2 + x3^2 - x2*x4 + x3*x4 - y1^2 + y0*y2
             - y3^2 + y2*y4 - y3*y4 - z1^2 + z0*z2 - z3^2 + z2*z4 - z3*z4 - x2,
       -x1*x2 - x0*x3 - x3*x4 - x4^2
            + y1*y2 + y0*y3 + y3*y4 + y4^2 + z1*z2 + z0*z3 + z3*z4 + z4^2 - x3,
       x2^2 - x1*x3 - x0*x4 + x4^2 - y2^2
                     + y1*y3 + y0*y4 - y4^2 - z2^2 + z1*z3 + z0*z4 - z4^2 - x4,
       -x0*y0 + x4*y1 + x3*y2 + x2*y3
                          + x1*y4 - y0*z0 + y4*z1 + y3*z2 + y2*z3 + y1*z4 - y0,
       -x1*y0 - x0*y1 - x4*y1 - x3*y2 + x4*y2 - x2*y3 + x3*y3
                          - x1*y4 + x2*y4 - y1*z0 - y0*z1 - y4*z1 - y3*z2
                                  + y4*z2 - y2*z3 + y3*z3 - y1*z4 + y2*z4 - y1,
       -x2*y0 - x1*y1 - x0*y2 - x4*y2 - x3*y3 + x4*y3 - x2*y4 + x3*y4
          - y2*z0 - y1*z1 - y0*z2 - y4*z2 - y3*z3 + y4*z3 - y2*z4 + y3*z4 - y2,
       -x3*y0 - x2*y1 - x1*y2 - x0*y3 - x4*y3 - x3*y4 + x4*y4
                  - y3*z0 - y2*z1 - y1*z2 - y0*z3 - y4*z3 - y3*z4 + y4*z4 - y3,
       -x4*y0 - x3*y1 - x2*y2 - x1*y3 - x0*y4 - x4*y4
                          - y4*z0 - y3*z1 - y2*z2 - y1*z3 - y0*z4 - y4*z4 - y4)
 of Multivariate Polynomial Ring in x0, x1, x2, x3, x4, y0, y1, y2, y3, y4,
 z0, z1, z2, z3, z4 over Finite Field of size 3
sage: J += sage.rings.ideal.FieldIdeal(J.ring())  # ensure radical ideal
sage: from sage.doctest.fixtures import reproducible_repr
sage: print(reproducible_repr(J.variety()))
[{x0: 1, x1: 0, x2: 0, x3: 0, x4: 0,
  y0: 0, y1: 0, y2: 0, y3: 0, y4: 0,
  z0: 0, z1: 0, z2: 0, z3: 0, z4: 0}]

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> k = GF(Integer(2)**Integer(2), names=('a',)); (a,) = k._first_ngens(1)
>>> P = PolynomialRing(k, Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> I = Ideal([x*y + Integer(1), a*x + Integer(1)])
>>> I.variety()
[{y: a, x: a + 1}]
>>> J = I.weil_restriction()
>>> J
Ideal (x0*y0 + x1*y1 + 1, x1*y0 + x0*y1 + x1*y1, x1 + 1, x0 + x1) of
 Multivariate Polynomial Ring in x0, x1, y0, y1 over Finite Field of size 2
>>> J += sage.rings.ideal.FieldIdeal(J.ring())  # ensure radical ideal
>>> J.variety()
[{y1: 1, y0: 0, x1: 1, x0: 1}]
>>> J.weil_restriction()  # returns J
Ideal (x0*y0 + x1*y1 + 1, x1*y0 + x0*y1 + x1*y1, x1 + 1, x0 + x1,
 x0^2 + x0, x1^2 + x1, y0^2 + y0, y1^2 + y1) of Multivariate
 Polynomial Ring in x0, x1, y0, y1 over Finite Field of size 2

>>> # needs sage.rings.finite_rings
>>> k = GF(Integer(3)**Integer(5), names=('a',)); (a,) = k._first_ngens(1)
>>> P = PolynomialRing(k, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> I = sage.rings.ideal.Katsura(P)
>>> I.dimension()
0
>>> I.variety()
 [{z: 0, y: 0, x: 1}]
>>> J = I.weil_restriction(); J
Ideal (x0 - y0 - z0 - 1,
       x1 - y1 - z1, x2 - y2 - z2, x3 - y3 - z3, x4 - y4 - z4,
       x0^2 + x2*x3 + x1*x4 - y0^2 - y2*y3 - y1*y4 - z0^2 - z2*z3 - z1*z4 - x0,
       -x0*x1 - x2*x3 - x3^2 - x1*x4 + x2*x4 + y0*y1 + y2*y3
            + y3^2 + y1*y4 - y2*y4 + z0*z1 + z2*z3 + z3^2 + z1*z4 - z2*z4 - x1,
       x1^2 - x0*x2 + x3^2 - x2*x4 + x3*x4 - y1^2 + y0*y2
             - y3^2 + y2*y4 - y3*y4 - z1^2 + z0*z2 - z3^2 + z2*z4 - z3*z4 - x2,
       -x1*x2 - x0*x3 - x3*x4 - x4^2
            + y1*y2 + y0*y3 + y3*y4 + y4^2 + z1*z2 + z0*z3 + z3*z4 + z4^2 - x3,
       x2^2 - x1*x3 - x0*x4 + x4^2 - y2^2
                     + y1*y3 + y0*y4 - y4^2 - z2^2 + z1*z3 + z0*z4 - z4^2 - x4,
       -x0*y0 + x4*y1 + x3*y2 + x2*y3
                          + x1*y4 - y0*z0 + y4*z1 + y3*z2 + y2*z3 + y1*z4 - y0,
       -x1*y0 - x0*y1 - x4*y1 - x3*y2 + x4*y2 - x2*y3 + x3*y3
                          - x1*y4 + x2*y4 - y1*z0 - y0*z1 - y4*z1 - y3*z2
                                  + y4*z2 - y2*z3 + y3*z3 - y1*z4 + y2*z4 - y1,
       -x2*y0 - x1*y1 - x0*y2 - x4*y2 - x3*y3 + x4*y3 - x2*y4 + x3*y4
          - y2*z0 - y1*z1 - y0*z2 - y4*z2 - y3*z3 + y4*z3 - y2*z4 + y3*z4 - y2,
       -x3*y0 - x2*y1 - x1*y2 - x0*y3 - x4*y3 - x3*y4 + x4*y4
                  - y3*z0 - y2*z1 - y1*z2 - y0*z3 - y4*z3 - y3*z4 + y4*z4 - y3,
       -x4*y0 - x3*y1 - x2*y2 - x1*y3 - x0*y4 - x4*y4
                          - y4*z0 - y3*z1 - y2*z2 - y1*z3 - y0*z4 - y4*z4 - y4)
 of Multivariate Polynomial Ring in x0, x1, x2, x3, x4, y0, y1, y2, y3, y4,
 z0, z1, z2, z3, z4 over Finite Field of size 3
>>> J += sage.rings.ideal.FieldIdeal(J.ring())  # ensure radical ideal
>>> from sage.doctest.fixtures import reproducible_repr
>>> print(reproducible_repr(J.variety()))
[{x0: 1, x1: 0, x2: 0, x3: 0, x4: 0,
  y0: 0, y1: 0, y2: 0, y3: 0, y4: 0,
  z0: 0, z1: 0, z2: 0, z3: 0, z4: 0}]

Weil restrictions are often used to study elliptic curves over extension fields so we give a simple example involving those:

sage: K.<a> = QuadraticField(1/3)                                           # needs sage.rings.number_field
sage: E = EllipticCurve(K, [1,2,3,4,5])                                     # needs sage.rings.number_field

>>> from sage.all import *
>>> K = QuadraticField(Integer(1)/Integer(3), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> E = EllipticCurve(K, [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)])                                     # needs sage.rings.number_field

We pick a point on E:

sage: p = E.lift_x(1); p                                                    # needs sage.rings.number_field
(1 : -6 : 1)

sage: I = E.defining_ideal(); I                                             # needs sage.rings.number_field
Ideal (-x^3 - 2*x^2*z + x*y*z + y^2*z - 4*x*z^2 + 3*y*z^2 - 5*z^3)
 of Multivariate Polynomial Ring in x, y, z
  over Number Field in a with defining polynomial x^2 - 1/3
   with a = 0.5773502691896258?

>>> from sage.all import *
>>> p = E.lift_x(Integer(1)); p                                                    # needs sage.rings.number_field
(1 : -6 : 1)

>>> I = E.defining_ideal(); I                                             # needs sage.rings.number_field
Ideal (-x^3 - 2*x^2*z + x*y*z + y^2*z - 4*x*z^2 + 3*y*z^2 - 5*z^3)
 of Multivariate Polynomial Ring in x, y, z
  over Number Field in a with defining polynomial x^2 - 1/3
   with a = 0.5773502691896258?

Of course, the point p is a root of all generators of I:

sage: I.subs(x=1, y=2, z=1)                                                 # needs sage.rings.number_field
Ideal (0) of Multivariate Polynomial Ring in x, y, z
 over Number Field in a with defining polynomial x^2 - 1/3
  with a = 0.5773502691896258?

>>> from sage.all import *
>>> I.subs(x=Integer(1), y=Integer(2), z=Integer(1))                                                 # needs sage.rings.number_field
Ideal (0) of Multivariate Polynomial Ring in x, y, z
 over Number Field in a with defining polynomial x^2 - 1/3
  with a = 0.5773502691896258?

I is also radical:

sage: I.radical() == I                                                      # needs sage.rings.number_field
True

>>> from sage.all import *
>>> I.radical() == I                                                      # needs sage.rings.number_field
True

So we compute its Weil restriction:

sage: J = I.weil_restriction(); J                                           # needs sage.rings.number_field
Ideal (-x0^3 - x0*x1^2 - 2*x0^2*z0 - 2/3*x1^2*z0 + x0*y0*z0 + y0^2*z0
         + 1/3*x1*y1*z0 + 1/3*y1^2*z0 - 4*x0*z0^2 + 3*y0*z0^2 - 5*z0^3
         - 4/3*x0*x1*z1 + 1/3*x1*y0*z1 + 1/3*x0*y1*z1 + 2/3*y0*y1*z1
         - 8/3*x1*z0*z1 + 2*y1*z0*z1 - 4/3*x0*z1^2 + y0*z1^2 - 5*z0*z1^2,
       -3*x0^2*x1 - 1/3*x1^3 - 4*x0*x1*z0 + x1*y0*z0 + x0*y1*z0
         + 2*y0*y1*z0 - 4*x1*z0^2 + 3*y1*z0^2 - 2*x0^2*z1 - 2/3*x1^2*z1
         + x0*y0*z1 + y0^2*z1 + 1/3*x1*y1*z1 + 1/3*y1^2*z1 - 8*x0*z0*z1
         + 6*y0*z0*z1 - 15*z0^2*z1 - 4/3*x1*z1^2 + y1*z1^2 - 5/3*z1^3)
of Multivariate Polynomial Ring in x0, x1, y0, y1, z0, z1 over Rational Field

>>> from sage.all import *
>>> J = I.weil_restriction(); J                                           # needs sage.rings.number_field
Ideal (-x0^3 - x0*x1^2 - 2*x0^2*z0 - 2/3*x1^2*z0 + x0*y0*z0 + y0^2*z0
         + 1/3*x1*y1*z0 + 1/3*y1^2*z0 - 4*x0*z0^2 + 3*y0*z0^2 - 5*z0^3
         - 4/3*x0*x1*z1 + 1/3*x1*y0*z1 + 1/3*x0*y1*z1 + 2/3*y0*y1*z1
         - 8/3*x1*z0*z1 + 2*y1*z0*z1 - 4/3*x0*z1^2 + y0*z1^2 - 5*z0*z1^2,
       -3*x0^2*x1 - 1/3*x1^3 - 4*x0*x1*z0 + x1*y0*z0 + x0*y1*z0
         + 2*y0*y1*z0 - 4*x1*z0^2 + 3*y1*z0^2 - 2*x0^2*z1 - 2/3*x1^2*z1
         + x0*y0*z1 + y0^2*z1 + 1/3*x1*y1*z1 + 1/3*y1^2*z1 - 8*x0*z0*z1
         + 6*y0*z0*z1 - 15*z0^2*z1 - 4/3*x1*z1^2 + y1*z1^2 - 5/3*z1^3)
of Multivariate Polynomial Ring in x0, x1, y0, y1, z0, z1 over Rational Field

We can check that the point p is still a root of all generators of J:

sage: J.subs(x0=1, y0=2, z0=1, x1=0, y1=0, z1=0)                            # needs sage.rings.number_field
Ideal (0, 0) of Multivariate Polynomial Ring in x0, x1, y0, y1, z0, z1
 over Rational Field

>>> from sage.all import *
>>> J.subs(x0=Integer(1), y0=Integer(2), z0=Integer(1), x1=Integer(0), y1=Integer(0), z1=Integer(0))                            # needs sage.rings.number_field
Ideal (0, 0) of Multivariate Polynomial Ring in x0, x1, y0, y1, z0, z1
 over Rational Field

Example for relative number fields:

sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: K.<w> = NumberField(x^5 - 2)
sage: R.<x> = K[]
sage: L.<v> = K.extension(x^2 + 1)
sage: S.<x,y> = L[]
sage: I = S.ideal([y^2 - x^3 - 1])
sage: I.weil_restriction()
Ideal (-x0^3 + 3*x0*x1^2 + y0^2 - y1^2 - 1, -3*x0^2*x1 + x1^3 + 2*y0*y1) of
 Multivariate Polynomial Ring in x0, x1, y0, y1
  over Number Field in w with defining polynomial x^5 - 2

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(5) - Integer(2), names=('w',)); (w,) = K._first_ngens(1)
>>> R = K['x']; (x,) = R._first_ngens(1)
>>> L = K.extension(x**Integer(2) + Integer(1), names=('v',)); (v,) = L._first_ngens(1)
>>> S = L['x, y']; (x, y,) = S._first_ngens(2)
>>> I = S.ideal([y**Integer(2) - x**Integer(3) - Integer(1)])
>>> I.weil_restriction()
Ideal (-x0^3 + 3*x0*x1^2 + y0^2 - y1^2 - 1, -3*x0^2*x1 + x1^3 + 2*y0*y1) of
 Multivariate Polynomial Ring in x0, x1, y0, y1
  over Number Field in w with defining polynomial x^5 - 2

Note

Based on a Singular implementation by Michael Brickenstein

class sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_macaulay2_repr[source]

Bases: object

An ideal in a multivariate polynomial ring, which has an underlying Macaulay2 ring associated to it.

EXAMPLES:

sage: R.<x,y,z,w> = PolynomialRing(ZZ, 4)
sage: I = ideal(x*y-z^2, y^2-w^2)
sage: I
Ideal (x*y - z^2, y^2 - w^2) of Multivariate Polynomial Ring in x, y, z, w over Integer Ring

>>> from sage.all import *
>>> R = PolynomialRing(ZZ, Integer(4), names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = R._first_ngens(4)
>>> I = ideal(x*y-z**Integer(2), y**Integer(2)-w**Integer(2))
>>> I
Ideal (x*y - z^2, y^2 - w^2) of Multivariate Polynomial Ring in x, y, z, w over Integer Ring
class sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_magma_repr[source]

Bases: object

class sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_quotient(ring, gens, coerce=True)[source]

Bases: MPolynomialIdeal

An ideal in a quotient of a multivariate polynomial ring.

EXAMPLES:

sage: Q.<x,y,z,w> = QQ['x,y,z,w'].quotient(['x*y-z^2', 'y^2-w^2'])
sage: I = ideal(x + y^2 + z - 1)
sage: I
Ideal (w^2 + x + z - 1) of Quotient
 of Multivariate Polynomial Ring in x, y, z, w over Rational Field
 by the ideal (x*y - z^2, y^2 - w^2)

>>> from sage.all import *
>>> Q = QQ['x,y,z,w'].quotient(['x*y-z^2', 'y^2-w^2'], names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = Q._first_ngens(4)
>>> I = ideal(x + y**Integer(2) + z - Integer(1))
>>> I
Ideal (w^2 + x + z - 1) of Quotient
 of Multivariate Polynomial Ring in x, y, z, w over Rational Field
 by the ideal (x*y - z^2, y^2 - w^2)
reduce(f)[source]

Reduce an element modulo a Gröbner basis for this ideal. This returns 0 if and only if the element is in this ideal. In any case, this reduction is unique up to monomial orders.

EXAMPLES:

sage: R.<T,U,V,W,X,Y,Z> = PolynomialRing(QQ, order='lex')
sage: I = R.ideal([T^2 + U^2 - 1, V^2 + W^2 - 1, X^2 + Y^2 + Z^2 - 1])
sage: Q.<t,u,v,w,x,y,z> = R.quotient(I)
sage: J = Q.ideal([u*v - x, u*w - y, t - z])
sage: J.reduce(t^2 - z^2)
0
sage: J.reduce(u^2)
-z^2 + 1
sage: t^2 - z^2 in J
True
sage: u^2 in J
False

>>> from sage.all import *
>>> R = PolynomialRing(QQ, order='lex', names=('T', 'U', 'V', 'W', 'X', 'Y', 'Z',)); (T, U, V, W, X, Y, Z,) = R._first_ngens(7)
>>> I = R.ideal([T**Integer(2) + U**Integer(2) - Integer(1), V**Integer(2) + W**Integer(2) - Integer(1), X**Integer(2) + Y**Integer(2) + Z**Integer(2) - Integer(1)])
>>> Q = R.quotient(I, names=('t', 'u', 'v', 'w', 'x', 'y', 'z',)); (t, u, v, w, x, y, z,) = Q._first_ngens(7)
>>> J = Q.ideal([u*v - x, u*w - y, t - z])
>>> J.reduce(t**Integer(2) - z**Integer(2))
0
>>> J.reduce(u**Integer(2))
-z^2 + 1
>>> t**Integer(2) - z**Integer(2) in J
True
>>> u**Integer(2) in J
False
class sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_singular_base_repr[source]

Bases: object

syzygy_module()[source]

Compute the first syzygy (i.e., the module of relations of the given generators) of the ideal.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ)
sage: f = 2*x^2 + y
sage: g = y
sage: h = 2*f + g
sage: I = Ideal([f,g,h])
sage: M = I.syzygy_module(); M
[       -2        -1         1]
[       -y 2*x^2 + y         0]
sage: G = vector(I.gens())
sage: M*G
(0, 0)

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(2)*x**Integer(2) + y
>>> g = y
>>> h = Integer(2)*f + g
>>> I = Ideal([f,g,h])
>>> M = I.syzygy_module(); M
[       -2        -1         1]
[       -y 2*x^2 + y         0]
>>> G = vector(I.gens())
>>> M*G
(0, 0)

ALGORITHM: Uses Singular’s syz command

class sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_singular_repr[source]

Bases: MPolynomialIdeal_singular_base_repr

An ideal in a multivariate polynomial ring, which has an underlying Singular ring associated to it.

associated_primes(algorithm='sy')[source]

Return a list of the associated primes of primary ideals of which the intersection is \(I\) = self.

An ideal \(Q\) is called primary if it is a proper ideal of the ring \(R\) and if whenever \(ab \in Q\) and \(a \not\in Q\) then \(b^n \in Q\) for some \(n \in \ZZ\).

If \(Q\) is a primary ideal of the ring \(R\), then the radical ideal \(P\) of \(Q\), i.e. \(P = \{a \in R, a^n \in Q\}\) for some \(n \in \ZZ\), is called the associated prime of \(Q\).

If \(I\) is a proper ideal of the ring \(R\) then there exists a decomposition in primary ideals \(Q_i\) such that

  • their intersection is \(I\)

  • none of the \(Q_i\) contains the intersection of the rest, and

  • the associated prime ideals of \(Q_i\) are pairwise different.

This method returns the associated primes of the \(Q_i\).

INPUT:

  • algorithm – string; one of

    • 'sy' – (default) use the Shimoyama-Yokoyama algorithm

    • 'gtz' – use the Gianni-Trager-Zacharias algorithm

OUTPUT: list of associated primes

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
sage: p = z^2 + 1; q = z^3 + 2
sage: I = (p*q^2, y - z^2) * R
sage: pd = I.associated_primes(); sorted(pd, key=str)
[Ideal (z^2 + 1, y + 1) of Multivariate Polynomial Ring in x, y, z over Rational Field,
 Ideal (z^3 + 2, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field]

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(3), order='lex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> p = z**Integer(2) + Integer(1); q = z**Integer(3) + Integer(2)
>>> I = (p*q**Integer(2), y - z**Integer(2)) * R
>>> pd = I.associated_primes(); sorted(pd, key=str)
[Ideal (z^2 + 1, y + 1) of Multivariate Polynomial Ring in x, y, z over Rational Field,
 Ideal (z^3 + 2, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field]

ALGORITHM:

Uses Singular.

REFERENCES:

  • Thomas Becker and Volker Weispfenning. Groebner Bases - A Computational Approach To Commutative Algebra. Springer, New York 1993.

basis_is_groebner(singular=None)[source]

Return True if the generators of this ideal (self.gens()) form a Groebner basis.

Let \(I\) be the set of generators of this ideal. The check is performed by trying to lift \(Syz(LM(I))\) to \(Syz(I)\) as \(I\) forms a Groebner basis if and only if for every element \(S\) in \(Syz(LM(I))\):

\[S * G = \sum_{i=0}^{m} h_ig_i ---->_G 0.\]

ALGORITHM:

Uses Singular.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: R.<a,b,c,d,e,f,g,h,i,j> = PolynomialRing(GF(127), 10)
sage: I = sage.rings.ideal.Cyclic(R, 4)
sage: I.basis_is_groebner()
False
sage: I2 = Ideal(I.groebner_basis())
sage: I2.basis_is_groebner()
True

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> R = PolynomialRing(GF(Integer(127)), Integer(10), names=('a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j',)); (a, b, c, d, e, f, g, h, i, j,) = R._first_ngens(10)
>>> I = sage.rings.ideal.Cyclic(R, Integer(4))
>>> I.basis_is_groebner()
False
>>> I2 = Ideal(I.groebner_basis())
>>> I2.basis_is_groebner()
True

A more complicated example:

sage: R.<U6,U5,U4,U3,U2, u6,u5,u4,u3,u2, h> = PolynomialRing(GF(7583))      # needs sage.rings.finite_rings
sage: l = [u6 + u5 + u4 + u3 + u2 - 3791*h,                                 # needs sage.rings.finite_rings
....:      U6 + U5 + U4 + U3 + U2 - 3791*h,
....:      U2*u2 - h^2, U3*u3 - h^2, U4*u4 - h^2,
....:      U5*u4 + U5*u3 + U4*u3 + U5*u2 + U4*u2 + U3*u2 - 3791*U5*h
....:        - 3791*U4*h - 3791*U3*h - 3791*U2*h - 2842*h^2,
....:      U4*u5 + U3*u5 + U2*u5 + U3*u4 + U2*u4 + U2*u3 - 3791*u5*h
....:        - 3791*u4*h - 3791*u3*h - 3791*u2*h - 2842*h^2,
....:      U5*u5 - h^2, U4*U2*u3 + U5*U3*u2 + U4*U3*u2 + U3^2*u2 - 3791*U5*U3*h
....:        - 3791*U4*U3*h - 3791*U3^2*h - 3791*U5*U2*h- 3791*U4*U2*h + U3*U2*h
....:        - 3791*U2^2*h - 3791*U4*u3*h - 3791*U4*u2*h - 3791*U3*u2*h
....:        - 2843*U5*h^2 + 1897*U4*h^2 - 946*U3*h^2 - 947*U2*h^2 + 2370*h^3,
....:      U3*u5*u4 + U2*u5*u4 + U3*u4^2 + U2*u4^2 + U2*u4*u3 - 3791*u5*u4*h
....:        - 3791*u4^2*h - 3791*u4*u3*h - 3791*u4*u2*h + u5*h^2 - 2842*u4*h^2,
....:      U2*u5*u4*u3 + U2*u4^2*u3 + U2*u4*u3^2 - 3791*u5*u4*u3*h
....:        - 3791*u4^2*u3*h - 3791*u4*u3^2*h - 3791*u4*u3*u2*h + u5*u4*h^2
....:        + u4^2*h^2 + u5*u3*h^2 - 2842*u4*u3*h^2,
....:      U5^2*U4*u3 + U5*U4^2*u3 + U5^2*U4*u2 + U5*U4^2*u2 + U5^2*U3*u2
....:        + 2*U5*U4*U3*u2 + U5*U3^2*u2 - 3791*U5^2*U4*h - 3791*U5*U4^2*h
....:        - 3791*U5^2*U3*h + U5*U4*U3*h - 3791*U5*U3^2*h - 3791*U5^2*U2*h
....:        + U5*U4*U2*h+ U5*U3*U2*h - 3791*U5*U2^2*h - 3791*U5*U3*u2*h
....:        - 2842*U5^2*h^2 + 1897*U5*U4*h^2 - U4^2*h^2 - 947*U5*U3*h^2
....:        - U4*U3*h^2 - 948*U5*U2*h^2 - U4*U2*h^2 - 1422*U5*h^3 + 3791*U4*h^3,
....:      u5*u4*u3*u2*h + u4^2*u3*u2*h + u4*u3^2*u2*h + u4*u3*u2^2*h
....:        + 2*u5*u4*u3*h^2 + 2*u4^2*u3*h^2 + 2*u4*u3^2*h^2 + 2*u5*u4*u2*h^2
....:        + 2*u4^2*u2*h^2 + 2*u5*u3*u2*h^2 + 1899*u4*u3*u2*h^2,
....:      U5^2*U4*U3*u2 + U5*U4^2*U3*u2 + U5*U4*U3^2*u2 - 3791*U5^2*U4*U3*h
....:        - 3791*U5*U4^2*U3*h - 3791*U5*U4*U3^2*h - 3791*U5*U4*U3*U2*h
....:        + 3791*U5*U4*U3*u2*h + U5^2*U4*h^2 + U5*U4^2*h^2 + U5^2*U3*h^2
....:        - U4^2*U3*h^2 - U5*U3^2*h^2 - U4*U3^2*h^2 - U5*U4*U2*h^2 - U5*U3*U2*h^2
....:        - U4*U3*U2*h^2 + 3791*U5*U4*h^3 + 3791*U5*U3*h^3 + 3791*U4*U3*h^3,
....:      u4^2*u3*u2*h^2 + 1515*u5*u3^2*u2*h^2 + u4*u3^2*u2*h^2
....:        + 1515*u5*u4*u2^2*h^2 + 1515*u5*u3*u2^2*h^2 + u4*u3*u2^2*h^2
....:        + 1521*u5*u4*u3*h^3 - 3028*u4^2*u3*h^3 - 3028*u4*u3^2*h^3
....:        + 1521*u5*u4*u2*h^3 - 3028*u4^2*u2*h^3 + 1521*u5*u3*u2*h^3
....:        + 3420*u4*u3*u2*h^3,
....:      U5^2*U4*U3*U2*h + U5*U4^2*U3*U2*h + U5*U4*U3^2*U2*h + U5*U4*U3*U2^2*h
....:        + 2*U5^2*U4*U3*h^2 + 2*U5*U4^2*U3*h^2 + 2*U5*U4*U3^2*h^2
....:        + 2*U5^2*U4*U2*h^2 + 2*U5*U4^2*U2*h^2 + 2*U5^2*U3*U2*h^2
....:        - 2*U4^2*U3*U2*h^2 - 2*U5*U3^2*U2*h^2 - 2*U4*U3^2*U2*h^2
....:        - 2*U5*U4*U2^2*h^2 - 2*U5*U3*U2^2*h^2 - 2*U4*U3*U2^2*h^2
....:        - U5*U4*U3*h^3 - U5*U4*U2*h^3 - U5*U3*U2*h^3 - U4*U3*U2*h^3]

sage: Ideal(l).basis_is_groebner()                                          # needs sage.rings.finite_rings
False
sage: gb = Ideal(l).groebner_basis()                                        # needs sage.rings.finite_rings
sage: Ideal(gb).basis_is_groebner()                                         # needs sage.rings.finite_rings
True

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(7583)), names=('U6', 'U5', 'U4', 'U3', 'U2', 'u6', 'u5', 'u4', 'u3', 'u2', 'h',)); (U6, U5, U4, U3, U2, u6, u5, u4, u3, u2, h,) = R._first_ngens(11)# needs sage.rings.finite_rings
>>> l = [u6 + u5 + u4 + u3 + u2 - Integer(3791)*h,                                 # needs sage.rings.finite_rings
...      U6 + U5 + U4 + U3 + U2 - Integer(3791)*h,
...      U2*u2 - h**Integer(2), U3*u3 - h**Integer(2), U4*u4 - h**Integer(2),
...      U5*u4 + U5*u3 + U4*u3 + U5*u2 + U4*u2 + U3*u2 - Integer(3791)*U5*h
...        - Integer(3791)*U4*h - Integer(3791)*U3*h - Integer(3791)*U2*h - Integer(2842)*h**Integer(2),
...      U4*u5 + U3*u5 + U2*u5 + U3*u4 + U2*u4 + U2*u3 - Integer(3791)*u5*h
...        - Integer(3791)*u4*h - Integer(3791)*u3*h - Integer(3791)*u2*h - Integer(2842)*h**Integer(2),
...      U5*u5 - h**Integer(2), U4*U2*u3 + U5*U3*u2 + U4*U3*u2 + U3**Integer(2)*u2 - Integer(3791)*U5*U3*h
...        - Integer(3791)*U4*U3*h - Integer(3791)*U3**Integer(2)*h - Integer(3791)*U5*U2*h- Integer(3791)*U4*U2*h + U3*U2*h
...        - Integer(3791)*U2**Integer(2)*h - Integer(3791)*U4*u3*h - Integer(3791)*U4*u2*h - Integer(3791)*U3*u2*h
...        - Integer(2843)*U5*h**Integer(2) + Integer(1897)*U4*h**Integer(2) - Integer(946)*U3*h**Integer(2) - Integer(947)*U2*h**Integer(2) + Integer(2370)*h**Integer(3),
...      U3*u5*u4 + U2*u5*u4 + U3*u4**Integer(2) + U2*u4**Integer(2) + U2*u4*u3 - Integer(3791)*u5*u4*h
...        - Integer(3791)*u4**Integer(2)*h - Integer(3791)*u4*u3*h - Integer(3791)*u4*u2*h + u5*h**Integer(2) - Integer(2842)*u4*h**Integer(2),
...      U2*u5*u4*u3 + U2*u4**Integer(2)*u3 + U2*u4*u3**Integer(2) - Integer(3791)*u5*u4*u3*h
...        - Integer(3791)*u4**Integer(2)*u3*h - Integer(3791)*u4*u3**Integer(2)*h - Integer(3791)*u4*u3*u2*h + u5*u4*h**Integer(2)
...        + u4**Integer(2)*h**Integer(2) + u5*u3*h**Integer(2) - Integer(2842)*u4*u3*h**Integer(2),
...      U5**Integer(2)*U4*u3 + U5*U4**Integer(2)*u3 + U5**Integer(2)*U4*u2 + U5*U4**Integer(2)*u2 + U5**Integer(2)*U3*u2
...        + Integer(2)*U5*U4*U3*u2 + U5*U3**Integer(2)*u2 - Integer(3791)*U5**Integer(2)*U4*h - Integer(3791)*U5*U4**Integer(2)*h
...        - Integer(3791)*U5**Integer(2)*U3*h + U5*U4*U3*h - Integer(3791)*U5*U3**Integer(2)*h - Integer(3791)*U5**Integer(2)*U2*h
...        + U5*U4*U2*h+ U5*U3*U2*h - Integer(3791)*U5*U2**Integer(2)*h - Integer(3791)*U5*U3*u2*h
...        - Integer(2842)*U5**Integer(2)*h**Integer(2) + Integer(1897)*U5*U4*h**Integer(2) - U4**Integer(2)*h**Integer(2) - Integer(947)*U5*U3*h**Integer(2)
...        - U4*U3*h**Integer(2) - Integer(948)*U5*U2*h**Integer(2) - U4*U2*h**Integer(2) - Integer(1422)*U5*h**Integer(3) + Integer(3791)*U4*h**Integer(3),
...      u5*u4*u3*u2*h + u4**Integer(2)*u3*u2*h + u4*u3**Integer(2)*u2*h + u4*u3*u2**Integer(2)*h
...        + Integer(2)*u5*u4*u3*h**Integer(2) + Integer(2)*u4**Integer(2)*u3*h**Integer(2) + Integer(2)*u4*u3**Integer(2)*h**Integer(2) + Integer(2)*u5*u4*u2*h**Integer(2)
...        + Integer(2)*u4**Integer(2)*u2*h**Integer(2) + Integer(2)*u5*u3*u2*h**Integer(2) + Integer(1899)*u4*u3*u2*h**Integer(2),
...      U5**Integer(2)*U4*U3*u2 + U5*U4**Integer(2)*U3*u2 + U5*U4*U3**Integer(2)*u2 - Integer(3791)*U5**Integer(2)*U4*U3*h
...        - Integer(3791)*U5*U4**Integer(2)*U3*h - Integer(3791)*U5*U4*U3**Integer(2)*h - Integer(3791)*U5*U4*U3*U2*h
...        + Integer(3791)*U5*U4*U3*u2*h + U5**Integer(2)*U4*h**Integer(2) + U5*U4**Integer(2)*h**Integer(2) + U5**Integer(2)*U3*h**Integer(2)
...        - U4**Integer(2)*U3*h**Integer(2) - U5*U3**Integer(2)*h**Integer(2) - U4*U3**Integer(2)*h**Integer(2) - U5*U4*U2*h**Integer(2) - U5*U3*U2*h**Integer(2)
...        - U4*U3*U2*h**Integer(2) + Integer(3791)*U5*U4*h**Integer(3) + Integer(3791)*U5*U3*h**Integer(3) + Integer(3791)*U4*U3*h**Integer(3),
...      u4**Integer(2)*u3*u2*h**Integer(2) + Integer(1515)*u5*u3**Integer(2)*u2*h**Integer(2) + u4*u3**Integer(2)*u2*h**Integer(2)
...        + Integer(1515)*u5*u4*u2**Integer(2)*h**Integer(2) + Integer(1515)*u5*u3*u2**Integer(2)*h**Integer(2) + u4*u3*u2**Integer(2)*h**Integer(2)
...        + Integer(1521)*u5*u4*u3*h**Integer(3) - Integer(3028)*u4**Integer(2)*u3*h**Integer(3) - Integer(3028)*u4*u3**Integer(2)*h**Integer(3)
...        + Integer(1521)*u5*u4*u2*h**Integer(3) - Integer(3028)*u4**Integer(2)*u2*h**Integer(3) + Integer(1521)*u5*u3*u2*h**Integer(3)
...        + Integer(3420)*u4*u3*u2*h**Integer(3),
...      U5**Integer(2)*U4*U3*U2*h + U5*U4**Integer(2)*U3*U2*h + U5*U4*U3**Integer(2)*U2*h + U5*U4*U3*U2**Integer(2)*h
...        + Integer(2)*U5**Integer(2)*U4*U3*h**Integer(2) + Integer(2)*U5*U4**Integer(2)*U3*h**Integer(2) + Integer(2)*U5*U4*U3**Integer(2)*h**Integer(2)
...        + Integer(2)*U5**Integer(2)*U4*U2*h**Integer(2) + Integer(2)*U5*U4**Integer(2)*U2*h**Integer(2) + Integer(2)*U5**Integer(2)*U3*U2*h**Integer(2)
...        - Integer(2)*U4**Integer(2)*U3*U2*h**Integer(2) - Integer(2)*U5*U3**Integer(2)*U2*h**Integer(2) - Integer(2)*U4*U3**Integer(2)*U2*h**Integer(2)
...        - Integer(2)*U5*U4*U2**Integer(2)*h**Integer(2) - Integer(2)*U5*U3*U2**Integer(2)*h**Integer(2) - Integer(2)*U4*U3*U2**Integer(2)*h**Integer(2)
...        - U5*U4*U3*h**Integer(3) - U5*U4*U2*h**Integer(3) - U5*U3*U2*h**Integer(3) - U4*U3*U2*h**Integer(3)]

>>> Ideal(l).basis_is_groebner()                                          # needs sage.rings.finite_rings
False
>>> gb = Ideal(l).groebner_basis()                                        # needs sage.rings.finite_rings
>>> Ideal(gb).basis_is_groebner()                                         # needs sage.rings.finite_rings
True

Note

From the Singular Manual for the reduce function we use in this method: ‘The result may have no meaning if the second argument (self) is not a standard basis’. I (malb) believe this refers to the mathematical fact that the results may have no meaning if self is no standard basis, i.e., Singular doesn’t ‘add’ any additional ‘nonsense’ to the result. So we may actually use reduce to determine if self is a Groebner basis.

complete_primary_decomposition()[source]

A decorator that creates a cached version of an instance method of a class.

Note

For proper behavior, the method must be a pure function (no side effects). Arguments to the method must be hashable or transformed into something hashable using key or they must define sage.structure.sage_object.SageObject._cache_key().

EXAMPLES:

sage: class Foo():
....:     @cached_method
....:     def f(self, t, x=2):
....:         print('computing')
....:         return t**x
sage: a = Foo()

>>> from sage.all import *
>>> class Foo():
...     @cached_method
...     def f(self, t, x=Integer(2)):
...         print('computing')
...         return t**x
>>> a = Foo()

The example shows that the actual computation takes place only once, and that the result is identical for equivalent input:

sage: res = a.f(3, 2); res
computing
9
sage: a.f(t = 3, x = 2) is res
True
sage: a.f(3) is res
True

>>> from sage.all import *
>>> res = a.f(Integer(3), Integer(2)); res
computing
9
>>> a.f(t = Integer(3), x = Integer(2)) is res
True
>>> a.f(Integer(3)) is res
True

Note, however, that the CachedMethod is replaced by a CachedMethodCaller or CachedMethodCallerNoArgs as soon as it is bound to an instance or class:

sage: P.<a,b,c,d> = QQ[]
sage: I = P*[a,b]
sage: type(I.__class__.gens)
<class 'sage.misc.cachefunc.CachedMethodCallerNoArgs'>

>>> from sage.all import *
>>> P = QQ['a, b, c, d']; (a, b, c, d,) = P._first_ngens(4)
>>> I = P*[a,b]
>>> type(I.__class__.gens)
<class 'sage.misc.cachefunc.CachedMethodCallerNoArgs'>

So, you would hardly ever see an instance of this class alive.

The parameter key can be used to pass a function which creates a custom cache key for inputs. In the following example, this parameter is used to ignore the algorithm keyword for caching:

sage: class A():
....:     def _f_normalize(self, x, algorithm): return x
....:     @cached_method(key=_f_normalize)
....:     def f(self, x, algorithm='default'): return x
sage: a = A()
sage: a.f(1, algorithm='default') is a.f(1) is a.f(1, algorithm='algorithm')
True

>>> from sage.all import *
>>> class A():
...     def _f_normalize(self, x, algorithm): return x
...     @cached_method(key=_f_normalize)
...     def f(self, x, algorithm='default'): return x
>>> a = A()
>>> a.f(Integer(1), algorithm='default') is a.f(Integer(1)) is a.f(Integer(1), algorithm='algorithm')
True

The parameter do_pickle can be used to enable pickling of the cache. Usually the cache is not stored when pickling:

sage: class A():
....:     @cached_method
....:     def f(self, x): return None
sage: import __main__
sage: __main__.A = A
sage: a = A()
sage: a.f(1)
sage: len(a.f.cache)
1
sage: b = loads(dumps(a))
sage: len(b.f.cache)
0

>>> from sage.all import *
>>> class A():
...     @cached_method
...     def f(self, x): return None
>>> import __main__
>>> __main__.A = A
>>> a = A()
>>> a.f(Integer(1))
>>> len(a.f.cache)
1
>>> b = loads(dumps(a))
>>> len(b.f.cache)
0

When do_pickle is set, the pickle contains the contents of the cache:

sage: class A():
....:     @cached_method(do_pickle=True)
....:     def f(self, x): return None
sage: __main__.A = A
sage: a = A()
sage: a.f(1)
sage: len(a.f.cache)
1
sage: b = loads(dumps(a))
sage: len(b.f.cache)
1

>>> from sage.all import *
>>> class A():
...     @cached_method(do_pickle=True)
...     def f(self, x): return None
>>> __main__.A = A
>>> a = A()
>>> a.f(Integer(1))
>>> len(a.f.cache)
1
>>> b = loads(dumps(a))
>>> len(b.f.cache)
1

Cached methods cannot be copied like usual methods, see Issue #12603. Copying them can lead to very surprising results:

sage: class A:
....:     @cached_method
....:     def f(self):
....:         return 1
sage: class B:
....:     g=A.f
....:     def f(self):
....:         return 2

sage: b=B()
sage: b.f()
2
sage: b.g()
1
sage: b.f()
1

>>> from sage.all import *
>>> class A:
...     @cached_method
...     def f(self):
...         return Integer(1)
>>> class B:
...     g=A.f
...     def f(self):
...         return Integer(2)

>>> b=B()
>>> b.f()
2
>>> b.g()
1
>>> b.f()
1
dimension(singular=None)[source]

The dimension of the ring modulo this ideal.

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(GF(32003), order='degrevlex')              # needs sage.rings.finite_rings
sage: I = ideal(x^2 - y, x^3)                                               # needs sage.rings.finite_rings
sage: I.dimension()                                                         # needs sage.rings.finite_rings
1

>>> from sage.all import *
>>> P = PolynomialRing(GF(Integer(32003)), order='degrevlex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)# needs sage.rings.finite_rings
>>> I = ideal(x**Integer(2) - y, x**Integer(3))                                               # needs sage.rings.finite_rings
>>> I.dimension()                                                         # needs sage.rings.finite_rings
1

If the ideal is the total ring, the dimension is \(-1\) by convention.

ALGORITHM:

For principal ideals, Theorem 3.5.1 of [Ger2008] is used. Otherwise Singular is used, unless the characteristic is too large. This requires computation of a Groebner basis, which can be very expensive.

For polynomials over a finite field of order too large for Singular, this falls back on a toy implementation of Buchberger to compute the Groebner basis, then uses the algorithm described in Chapter 9, Section 1 of Cox, Little, and O’Shea’s “Ideals, Varieties, and Algorithms”.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: R.<x,y> = PolynomialRing(GF(2147483659^2), order='lex')
sage: I = R.ideal([x*y, x*y + 1])
sage: I.dimension()
verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
-1
sage: I=ideal([x*(x*y+1), y*(x*y+1)])
sage: I.dimension()
verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
1
sage: I = R.ideal([x^3*y, x*y^2])
sage: I.dimension()
verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
1
sage: R.<x,y> = PolynomialRing(GF(2147483659^2), order='lex')
sage: I = R.ideal(0)
sage: I.dimension()
2

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> R = PolynomialRing(GF(Integer(2147483659)**Integer(2)), order='lex', names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> I = R.ideal([x*y, x*y + Integer(1)])
>>> I.dimension()
verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
-1
>>> I=ideal([x*(x*y+Integer(1)), y*(x*y+Integer(1))])
>>> I.dimension()
verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
1
>>> I = R.ideal([x**Integer(3)*y, x*y**Integer(2)])
>>> I.dimension()
verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
1
>>> R = PolynomialRing(GF(Integer(2147483659)**Integer(2)), order='lex', names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> I = R.ideal(Integer(0))
>>> I.dimension()
2
elimination_ideal(variables, algorithm=None, *args, **kwds)[source]

Return the elimination ideal of this ideal with respect to the variables given in variables.

INPUT:

  • variables – list or tuple of variables in self.ring()

  • algorithm – determines the algorithm to use, see below for available algorithms

ALGORITHMS:

  • 'libsingular:eliminate' – libSingular’s eliminate command (default)

  • 'giac:eliminate' – Giac’s eliminate command (if available)

If only a system is given - e.g. ‘giac’ - the default algorithm is chosen for that system.

EXAMPLES:

sage: R.<x,y,t,s,z> = PolynomialRing(QQ,5)
sage: I = R * [x - t, y - t^2, z - t^3, s - x + y^3]
sage: J = I.elimination_ideal([t, s]); J
Ideal (y^2 - x*z, x*y - z, x^2 - y)
 of Multivariate Polynomial Ring in x, y, t, s, z over Rational Field

>>> from sage.all import *
>>> R = PolynomialRing(QQ,Integer(5), names=('x', 'y', 't', 's', 'z',)); (x, y, t, s, z,) = R._first_ngens(5)
>>> I = R * [x - t, y - t**Integer(2), z - t**Integer(3), s - x + y**Integer(3)]
>>> J = I.elimination_ideal([t, s]); J
Ideal (y^2 - x*z, x*y - z, x^2 - y)
 of Multivariate Polynomial Ring in x, y, t, s, z over Rational Field

You can use Giac to compute the elimination ideal:

sage: # needs sage.libs.giac
sage: print("possible output from giac", flush=True); I.elimination_ideal([t, s], algorithm='giac') == J
possible output...
True

>>> from sage.all import *
>>> # needs sage.libs.giac
>>> print("possible output from giac", flush=True); I.elimination_ideal([t, s], algorithm='giac') == J
possible output...
True

The list of available Giac options is provided at sage.libs.giac.groebner_basis().

ALGORITHM:

Uses Singular, or Giac (if available).

Note

Requires computation of a Groebner basis, which can be a very expensive operation.

free_resolution(*args, **kwds)[source]

Return a free resolution of self.

For input options, see FreeResolution.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ)
sage: f = 2*x^2 + y
sage: g = y
sage: h = 2*f + g
sage: I = R.ideal([f,g,h])
sage: res = I.free_resolution()
sage: res
S^1 <-- S^2 <-- S^1 <-- 0
sage: ascii_art(res.chain_complex())
                            [-x^2]
            [  y x^2]       [   y]
 0 <-- C_0 <---------- C_1 <------- C_2 <-- 0

sage: q = ZZ['q'].fraction_field().gen()
sage: R.<x,y,z> = q.parent()[]
sage: I = R.ideal([x^2+y^2+z^2, x*y*z^4])
sage: I.free_resolution()
Traceback (most recent call last):
...
NotImplementedError: the ring must be a polynomial ring using Singular

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(2)*x**Integer(2) + y
>>> g = y
>>> h = Integer(2)*f + g
>>> I = R.ideal([f,g,h])
>>> res = I.free_resolution()
>>> res
S^1 <-- S^2 <-- S^1 <-- 0
>>> ascii_art(res.chain_complex())
                            [-x^2]
            [  y x^2]       [   y]
 0 <-- C_0 <---------- C_1 <------- C_2 <-- 0

>>> q = ZZ['q'].fraction_field().gen()
>>> R = q.parent()['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> I = R.ideal([x**Integer(2)+y**Integer(2)+z**Integer(2), x*y*z**Integer(4)])
>>> I.free_resolution()
Traceback (most recent call last):
...
NotImplementedError: the ring must be a polynomial ring using Singular
genus()[source]

A decorator that creates a cached version of an instance method of a class.

Note

For proper behavior, the method must be a pure function (no side effects). Arguments to the method must be hashable or transformed into something hashable using key or they must define sage.structure.sage_object.SageObject._cache_key().

EXAMPLES:

sage: class Foo():
....:     @cached_method
....:     def f(self, t, x=2):
....:         print('computing')
....:         return t**x
sage: a = Foo()

>>> from sage.all import *
>>> class Foo():
...     @cached_method
...     def f(self, t, x=Integer(2)):
...         print('computing')
...         return t**x
>>> a = Foo()

The example shows that the actual computation takes place only once, and that the result is identical for equivalent input:

sage: res = a.f(3, 2); res
computing
9
sage: a.f(t = 3, x = 2) is res
True
sage: a.f(3) is res
True

>>> from sage.all import *
>>> res = a.f(Integer(3), Integer(2)); res
computing
9
>>> a.f(t = Integer(3), x = Integer(2)) is res
True
>>> a.f(Integer(3)) is res
True

Note, however, that the CachedMethod is replaced by a CachedMethodCaller or CachedMethodCallerNoArgs as soon as it is bound to an instance or class:

sage: P.<a,b,c,d> = QQ[]
sage: I = P*[a,b]
sage: type(I.__class__.gens)
<class 'sage.misc.cachefunc.CachedMethodCallerNoArgs'>

>>> from sage.all import *
>>> P = QQ['a, b, c, d']; (a, b, c, d,) = P._first_ngens(4)
>>> I = P*[a,b]
>>> type(I.__class__.gens)
<class 'sage.misc.cachefunc.CachedMethodCallerNoArgs'>

So, you would hardly ever see an instance of this class alive.

The parameter key can be used to pass a function which creates a custom cache key for inputs. In the following example, this parameter is used to ignore the algorithm keyword for caching:

sage: class A():
....:     def _f_normalize(self, x, algorithm): return x
....:     @cached_method(key=_f_normalize)
....:     def f(self, x, algorithm='default'): return x
sage: a = A()
sage: a.f(1, algorithm='default') is a.f(1) is a.f(1, algorithm='algorithm')
True

>>> from sage.all import *
>>> class A():
...     def _f_normalize(self, x, algorithm): return x
...     @cached_method(key=_f_normalize)
...     def f(self, x, algorithm='default'): return x
>>> a = A()
>>> a.f(Integer(1), algorithm='default') is a.f(Integer(1)) is a.f(Integer(1), algorithm='algorithm')
True

The parameter do_pickle can be used to enable pickling of the cache. Usually the cache is not stored when pickling:

sage: class A():
....:     @cached_method
....:     def f(self, x): return None
sage: import __main__
sage: __main__.A = A
sage: a = A()
sage: a.f(1)
sage: len(a.f.cache)
1
sage: b = loads(dumps(a))
sage: len(b.f.cache)
0

>>> from sage.all import *
>>> class A():
...     @cached_method
...     def f(self, x): return None
>>> import __main__
>>> __main__.A = A
>>> a = A()
>>> a.f(Integer(1))
>>> len(a.f.cache)
1
>>> b = loads(dumps(a))
>>> len(b.f.cache)
0

When do_pickle is set, the pickle contains the contents of the cache:

sage: class A():
....:     @cached_method(do_pickle=True)
....:     def f(self, x): return None
sage: __main__.A = A
sage: a = A()
sage: a.f(1)
sage: len(a.f.cache)
1
sage: b = loads(dumps(a))
sage: len(b.f.cache)
1

>>> from sage.all import *
>>> class A():
...     @cached_method(do_pickle=True)
...     def f(self, x): return None
>>> __main__.A = A
>>> a = A()
>>> a.f(Integer(1))
>>> len(a.f.cache)
1
>>> b = loads(dumps(a))
>>> len(b.f.cache)
1

Cached methods cannot be copied like usual methods, see Issue #12603. Copying them can lead to very surprising results:

sage: class A:
....:     @cached_method
....:     def f(self):
....:         return 1
sage: class B:
....:     g=A.f
....:     def f(self):
....:         return 2

sage: b=B()
sage: b.f()
2
sage: b.g()
1
sage: b.f()
1

>>> from sage.all import *
>>> class A:
...     @cached_method
...     def f(self):
...         return Integer(1)
>>> class B:
...     g=A.f
...     def f(self):
...         return Integer(2)

>>> b=B()
>>> b.f()
2
>>> b.g()
1
>>> b.f()
1
graded_free_resolution(*args, **kwds)[source]

Return a graded free resolution of self.

For input options, see GradedFreeResolution.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ)
sage: f = 2*x^2 + y^2
sage: g = y^2
sage: h = 2*f + g
sage: I = R.ideal([f,g,h])
sage: I.graded_free_resolution(shifts=[1])
S(-1) <-- S(-3)⊕S(-3) <-- S(-5) <-- 0

sage: f = 2*x^2 + y
sage: g = y
sage: h = 2*f + g
sage: I = R.ideal([f,g,h])
sage: I.graded_free_resolution(degrees=[1,2])
S(0) <-- S(-2)⊕S(-2) <-- S(-4) <-- 0

sage: q = ZZ['q'].fraction_field().gen()
sage: R.<x,y,z> = q.parent()[]
sage: I = R.ideal([x^2+y^2+z^2, x*y*z^4])
sage: I.graded_free_resolution()
Traceback (most recent call last):
...
NotImplementedError: the ring must be a polynomial ring using Singular

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(2)*x**Integer(2) + y**Integer(2)
>>> g = y**Integer(2)
>>> h = Integer(2)*f + g
>>> I = R.ideal([f,g,h])
>>> I.graded_free_resolution(shifts=[Integer(1)])
S(-1) <-- S(-3)⊕S(-3) <-- S(-5) <-- 0

>>> f = Integer(2)*x**Integer(2) + y
>>> g = y
>>> h = Integer(2)*f + g
>>> I = R.ideal([f,g,h])
>>> I.graded_free_resolution(degrees=[Integer(1),Integer(2)])
S(0) <-- S(-2)⊕S(-2) <-- S(-4) <-- 0

>>> q = ZZ['q'].fraction_field().gen()
>>> R = q.parent()['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> I = R.ideal([x**Integer(2)+y**Integer(2)+z**Integer(2), x*y*z**Integer(4)])
>>> I.graded_free_resolution()
Traceback (most recent call last):
...
NotImplementedError: the ring must be a polynomial ring using Singular
hilbert_numerator(grading=None, algorithm='sage')[source]

Return the Hilbert numerator of this ideal.

INPUT:

  • grading – (optional) a list or tuple of integers

  • algorithm – (default: 'sage') must be either 'sage' or 'singular'

Let \(I\) (which is self) be a homogeneous ideal and \(R = \bigoplus_d R_d\) (which is self.ring()) be a graded commutative algebra over a field \(K\). Then the Hilbert function is defined as \(H(d) = \dim_K R_d\) and the Hilbert series of \(I\) is defined as the formal power series \(HS(t) = \sum_{d=0}^{\infty} H(d) t^d\).

This power series can be expressed as \(HS(t) = Q(t) / (1-t)^n\) where \(Q(t)\) is a polynomial over \(\ZZ\) and \(n\) the number of variables in \(R\). This method returns \(Q(t)\), the numerator; hence the name, hilbert_numerator. An optional grading can be given, in which case the graded (or weighted) Hilbert numerator is given.

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ)
sage: I = Ideal([x^3*y^2 + 3*x^2*y^2*z + y^3*z^2 + z^5])
sage: I.hilbert_numerator()                                                 # needs sage.libs.flint
-t^5 + 1
sage: R.<a,b> = PolynomialRing(QQ)
sage: J = R.ideal([a^2*b, a*b^2])
sage: J.hilbert_numerator()                                                 # needs sage.libs.flint
t^4 - 2*t^3 + 1
sage: J.hilbert_numerator(grading=(10,3))                                   # needs sage.libs.flint
t^26 - t^23 - t^16 + 1

>>> from sage.all import *
>>> P = PolynomialRing(QQ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> I = Ideal([x**Integer(3)*y**Integer(2) + Integer(3)*x**Integer(2)*y**Integer(2)*z + y**Integer(3)*z**Integer(2) + z**Integer(5)])
>>> I.hilbert_numerator()                                                 # needs sage.libs.flint
-t^5 + 1
>>> R = PolynomialRing(QQ, names=('a', 'b',)); (a, b,) = R._first_ngens(2)
>>> J = R.ideal([a**Integer(2)*b, a*b**Integer(2)])
>>> J.hilbert_numerator()                                                 # needs sage.libs.flint
t^4 - 2*t^3 + 1
>>> J.hilbert_numerator(grading=(Integer(10),Integer(3)))                                   # needs sage.libs.flint
t^26 - t^23 - t^16 + 1
hilbert_polynomial(algorithm='sage')[source]

Return the Hilbert polynomial of this ideal.

INPUT:

  • algorithm – (default: 'sage') must be either 'sage' or 'singular'

Let \(I\) (which is self) be a homogeneous ideal and \(R = \bigoplus_d R_d\) (which is self.ring()) be a graded commutative algebra over a field \(K\). The Hilbert polynomial is the unique polynomial \(HP(t)\) with rational coefficients such that \(HP(d) = \dim_K R_d\) for all but finitely many positive integers \(d\).

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ)
sage: I = Ideal([x^3*y^2 + 3*x^2*y^2*z + y^3*z^2 + z^5])
sage: I.hilbert_polynomial()                                                # needs sage.libs.flint
5*t - 5

>>> from sage.all import *
>>> P = PolynomialRing(QQ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> I = Ideal([x**Integer(3)*y**Integer(2) + Integer(3)*x**Integer(2)*y**Integer(2)*z + y**Integer(3)*z**Integer(2) + z**Integer(5)])
>>> I.hilbert_polynomial()                                                # needs sage.libs.flint
5*t - 5

Of course, the Hilbert polynomial of a zero-dimensional ideal is zero:

sage: J0 = Ideal([x^3*y^2 + 3*x^2*y^2*z + y^3*z^2 + z^5,
....:             y^3 - 2*x*z^2 + x*y, x^4 + x*y - y*z^2])
sage: J = P*[m.lm() for m in J0.groebner_basis()]
sage: J.dimension()
0
sage: J.hilbert_polynomial()                                                # needs sage.libs.flint
0

>>> from sage.all import *
>>> J0 = Ideal([x**Integer(3)*y**Integer(2) + Integer(3)*x**Integer(2)*y**Integer(2)*z + y**Integer(3)*z**Integer(2) + z**Integer(5),
...             y**Integer(3) - Integer(2)*x*z**Integer(2) + x*y, x**Integer(4) + x*y - y*z**Integer(2)])
>>> J = P*[m.lm() for m in J0.groebner_basis()]
>>> J.dimension()
0
>>> J.hilbert_polynomial()                                                # needs sage.libs.flint
0

It is possible to request a computation using the Singular library:

sage: I.hilbert_polynomial(algorithm='singular') == I.hilbert_polynomial()  # needs sage.libs.flint
True
sage: J.hilbert_polynomial(algorithm='singular') == J.hilbert_polynomial()  # needs sage.libs.flint
True

>>> from sage.all import *
>>> I.hilbert_polynomial(algorithm='singular') == I.hilbert_polynomial()  # needs sage.libs.flint
True
>>> J.hilbert_polynomial(algorithm='singular') == J.hilbert_polynomial()  # needs sage.libs.flint
True

Here is a bigger examples:

sage: n = 4; m = 11; P = PolynomialRing(QQ, n * m, "x"); x = P.gens()
sage: M = Matrix(n, x)
sage: Minors = P.ideal(M.minors(2))
sage: hp = Minors.hilbert_polynomial(); hp                                  # needs sage.libs.flint
1/21772800*t^13 + 61/21772800*t^12 + 1661/21772800*t^11
 + 26681/21772800*t^10 + 93841/7257600*t^9 + 685421/7257600*t^8
 + 1524809/3110400*t^7 + 39780323/21772800*t^6 + 6638071/1360800*t^5
 + 12509761/1360800*t^4 + 2689031/226800*t^3 + 1494509/151200*t^2
 + 12001/2520*t + 1

>>> from sage.all import *
>>> n = Integer(4); m = Integer(11); P = PolynomialRing(QQ, n * m, "x"); x = P.gens()
>>> M = Matrix(n, x)
>>> Minors = P.ideal(M.minors(Integer(2)))
>>> hp = Minors.hilbert_polynomial(); hp                                  # needs sage.libs.flint
1/21772800*t^13 + 61/21772800*t^12 + 1661/21772800*t^11
 + 26681/21772800*t^10 + 93841/7257600*t^9 + 685421/7257600*t^8
 + 1524809/3110400*t^7 + 39780323/21772800*t^6 + 6638071/1360800*t^5
 + 12509761/1360800*t^4 + 2689031/226800*t^3 + 1494509/151200*t^2
 + 12001/2520*t + 1

Because Singular uses 32-bit integers, the above example would fail with Singular. We don’t test it here, as it has a side-effect on other tests that is not understood yet (see Issue #26300):

sage: Minors.hilbert_polynomial(algorithm='singular')       # not tested
Traceback (most recent call last):
...
RuntimeError: error in Singular function call 'hilbPoly':
int overflow in hilb 1
error occurred in or before poly.lib::hilbPoly line 58: `   intvec v=hilb(I,2);`
expected intvec-expression. type 'help intvec;'

>>> from sage.all import *
>>> Minors.hilbert_polynomial(algorithm='singular')       # not tested
Traceback (most recent call last):
...
RuntimeError: error in Singular function call 'hilbPoly':
int overflow in hilb 1
error occurred in or before poly.lib::hilbPoly line 58: `   intvec v=hilb(I,2);`
expected intvec-expression. type 'help intvec;'

Note that in this example, the Hilbert polynomial gives the coefficients of the Hilbert-Poincaré series in all degrees:

sage: P = PowerSeriesRing(QQ, 't', default_prec=50)
sage: hs = Minors.hilbert_series()                                          # needs sage.libs.flint
sage: list(P(hs.numerator()) / P(hs.denominator())) == [hp(t=k)             # needs sage.libs.flint
....:                                                   for k in range(50)]
True

>>> from sage.all import *
>>> P = PowerSeriesRing(QQ, 't', default_prec=Integer(50))
>>> hs = Minors.hilbert_series()                                          # needs sage.libs.flint
>>> list(P(hs.numerator()) / P(hs.denominator())) == [hp(t=k)             # needs sage.libs.flint
...                                                   for k in range(Integer(50))]
True
hilbert_series(grading=None, algorithm='sage')[source]

Return the Hilbert series of this ideal.

INPUT:

  • grading – (optional) a list or tuple of integers

  • algorithm – (default: 'sage') must be either 'sage' or 'singular'

Let \(I\) (which is self) be a homogeneous ideal and \(R = \bigoplus_d R_d\) (which is self.ring()) be a graded commutative algebra over a field \(K\). Then the Hilbert function is defined as \(H(d) = \dim_K R_d\) and the Hilbert series of \(I\) is defined as the formal power series \(HS(t) = \sum_{d=0}^{\infty} H(d) t^d\).

This power series can be expressed as \(HS(t) = Q(t) / (1-t)^n\) where \(Q(t)\) is a polynomial over \(\ZZ\) and \(n\) the number of variables in \(R\). This method returns \(Q(t) / (1-t)^n\), normalised so that the leading monomial of the numerator is positive.

An optional grading can be given, in which case the graded (or weighted) Hilbert series is given.

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ)
sage: I = Ideal([x^3*y^2 + 3*x^2*y^2*z + y^3*z^2 + z^5])
sage: I.hilbert_series()                                                    # needs sage.libs.flint
(t^4 + t^3 + t^2 + t + 1)/(t^2 - 2*t + 1)
sage: R.<a,b> = PolynomialRing(QQ)
sage: J = R.ideal([a^2*b, a*b^2])
sage: J.hilbert_series()                                                    # needs sage.libs.flint
(t^3 - t^2 - t - 1)/(t - 1)
sage: J.hilbert_series(grading=(10,3))                                      # needs sage.libs.flint
(t^25 + t^24 + t^23 - t^15 - t^14 - t^13 - t^12 - t^11
 - t^10 - t^9 - t^8 - t^7 - t^6 - t^5 - t^4 - t^3 - t^2
 - t - 1)/(t^12 + t^11 + t^10 - t^2 - t - 1)

sage: K = R.ideal([a^2*b^3, a*b^4 + a^3*b^2])
sage: K.hilbert_series(grading=[1,2])                                       # needs sage.libs.flint
(t^11 + t^8 - t^6 - t^5 - t^4 - t^3 - t^2 - t - 1)/(t^2 - 1)
sage: K.hilbert_series(grading=[2,1])                                       # needs sage.libs.flint
(2*t^7 - t^6 - t^4 - t^2 - 1)/(t - 1)

>>> from sage.all import *
>>> P = PolynomialRing(QQ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> I = Ideal([x**Integer(3)*y**Integer(2) + Integer(3)*x**Integer(2)*y**Integer(2)*z + y**Integer(3)*z**Integer(2) + z**Integer(5)])
>>> I.hilbert_series()                                                    # needs sage.libs.flint
(t^4 + t^3 + t^2 + t + 1)/(t^2 - 2*t + 1)
>>> R = PolynomialRing(QQ, names=('a', 'b',)); (a, b,) = R._first_ngens(2)
>>> J = R.ideal([a**Integer(2)*b, a*b**Integer(2)])
>>> J.hilbert_series()                                                    # needs sage.libs.flint
(t^3 - t^2 - t - 1)/(t - 1)
>>> J.hilbert_series(grading=(Integer(10),Integer(3)))                                      # needs sage.libs.flint
(t^25 + t^24 + t^23 - t^15 - t^14 - t^13 - t^12 - t^11
 - t^10 - t^9 - t^8 - t^7 - t^6 - t^5 - t^4 - t^3 - t^2
 - t - 1)/(t^12 + t^11 + t^10 - t^2 - t - 1)

>>> K = R.ideal([a**Integer(2)*b**Integer(3), a*b**Integer(4) + a**Integer(3)*b**Integer(2)])
>>> K.hilbert_series(grading=[Integer(1),Integer(2)])                                       # needs sage.libs.flint
(t^11 + t^8 - t^6 - t^5 - t^4 - t^3 - t^2 - t - 1)/(t^2 - 1)
>>> K.hilbert_series(grading=[Integer(2),Integer(1)])                                       # needs sage.libs.flint
(2*t^7 - t^6 - t^4 - t^2 - 1)/(t - 1)
integral_closure(p=0, r=True, singular=None)[source]

Let \(I\) = self.

Return the integral closure of \(I, ..., I^p\), where \(sI\) is an ideal in the polynomial ring \(R=k[x(1),...x(n)]\). If \(p\) is not given, or \(p=0\), compute the closure of all powers up to the maximum degree in \(t\) occurring in the closure of \(R[It]\) (so this is the last power whose closure is not just the sum/product of the smaller). If \(r\) is given and r is True, I.integral_closure() starts with a check whether \(I\) is already a radical ideal.

INPUT:

  • p – powers of \(I\) (default: 0)

  • r – check whether self is a radical ideal first (default: True)

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: I = ideal([x^2, x*y^4, y^5])
sage: I.integral_closure()
[x^2, x*y^4, y^5, x*y^3]

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> I = ideal([x**Integer(2), x*y**Integer(4), y**Integer(5)])
>>> I.integral_closure()
[x^2, x*y^4, y^5, x*y^3]

ALGORITHM:

Uses libSINGULAR.

interreduced_basis()[source]

If this ideal is spanned by \((f_1, ..., f_n)\), return \((g_1, ..., g_s)\) such that:

  • \((f_1,...,f_n) = (g_1,...,g_s)\)

  • \(LT(g_i) \neq LT(g_j)\) for all \(i \neq j\)

  • \(LT(g_i)\) does not divide \(m\) for all monomials \(m\) of \(\{g_1,...,g_{i-1},g_{i+1},...,g_s\}\)

  • \(LC(g_i) = 1\) for all \(i\) if the coefficient ring is a field.

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(QQ)
sage: I = Ideal([z*x + y^3, z + y^3, z + x*y])
sage: I.interreduced_basis()
[y^3 + z, x*y + z, x*z - z]

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> I = Ideal([z*x + y**Integer(3), z + y**Integer(3), z + x*y])
>>> I.interreduced_basis()
[y^3 + z, x*y + z, x*z - z]

Note that tail reduction for local orderings is not well-defined:

sage: R.<x,y,z> = PolynomialRing(QQ,order='negdegrevlex')
sage: I = Ideal([z*x + y^3, z + y^3, z + x*y])
sage: I.interreduced_basis()
[z + x*y, x*y - y^3, x^2*y - y^3]

>>> from sage.all import *
>>> R = PolynomialRing(QQ,order='negdegrevlex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> I = Ideal([z*x + y**Integer(3), z + y**Integer(3), z + x*y])
>>> I.interreduced_basis()
[z + x*y, x*y - y^3, x^2*y - y^3]

A fixed error with nonstandard base fields:

sage: R.<t> = QQ['t']
sage: K.<x,y> = R.fraction_field()['x,y']
sage: I = t*x * K
sage: I.interreduced_basis()
[x]

>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> K = R.fraction_field()['x,y']; (x, y,) = K._first_ngens(2)
>>> I = t*x * K
>>> I.interreduced_basis()
[x]

The interreduced basis of 0 is 0:

sage: P.<x,y,z> = GF(2)[]
sage: Ideal(P(0)).interreduced_basis()
[0]

>>> from sage.all import *
>>> P = GF(Integer(2))['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> Ideal(P(Integer(0))).interreduced_basis()
[0]

ALGORITHM:

Uses Singular’s interred command or sage.rings.polynomial.toy_buchberger.inter_reduction() if conversion to Singular fails.

intersection(*others)[source]

Return the intersection of the arguments with this ideal.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ, 2, order='lex')
sage: I = x*R
sage: J = y*R
sage: I.intersection(J)
Ideal (x*y) of Multivariate Polynomial Ring in x, y over Rational Field

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(2), order='lex', names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> I = x*R
>>> J = y*R
>>> I.intersection(J)
Ideal (x*y) of Multivariate Polynomial Ring in x, y over Rational Field

The following simple example illustrates that the product need not equal the intersection.

sage: I = (x^2, y) * R
sage: J = (y^2, x) * R
sage: K = I.intersection(J); K
Ideal (y^2, x*y, x^2) of Multivariate Polynomial Ring in x, y over Rational Field
sage: IJ = I*J; IJ
Ideal (x^2*y^2, x^3, y^3, x*y)
 of Multivariate Polynomial Ring in x, y over Rational Field
sage: IJ == K
False

>>> from sage.all import *
>>> I = (x**Integer(2), y) * R
>>> J = (y**Integer(2), x) * R
>>> K = I.intersection(J); K
Ideal (y^2, x*y, x^2) of Multivariate Polynomial Ring in x, y over Rational Field
>>> IJ = I*J; IJ
Ideal (x^2*y^2, x^3, y^3, x*y)
 of Multivariate Polynomial Ring in x, y over Rational Field
>>> IJ == K
False

Intersection of several ideals:

sage: R.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
sage: I1 = x * R
sage: I2 = y * R
sage: I3 = (x, y) * R
sage: I4 = (x^2 + x*y*z, y^2 - z^3*y, z^3 + y^5*x*z) * R
sage: I1.intersection(I2, I3, I4).groebner_basis()
[x^2*y + x*y*z^4, x*y^2 - x*y*z^3, x*y*z^20 - x*y*z^3]

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(3), order='lex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> I1 = x * R
>>> I2 = y * R
>>> I3 = (x, y) * R
>>> I4 = (x**Integer(2) + x*y*z, y**Integer(2) - z**Integer(3)*y, z**Integer(3) + y**Integer(5)*x*z) * R
>>> I1.intersection(I2, I3, I4).groebner_basis()
[x^2*y + x*y*z^4, x*y^2 - x*y*z^3, x*y*z^20 - x*y*z^3]

The ideals must share the same ring:

sage: R2.<x,y> = PolynomialRing(QQ, 2, order='lex')
sage: R3.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
sage: I2 = x*R2
sage: I3 = x*R3
sage: I2.intersection(I3)
Traceback (most recent call last):
...
TypeError: Intersection is only available for ideals of the same ring.

>>> from sage.all import *
>>> R2 = PolynomialRing(QQ, Integer(2), order='lex', names=('x', 'y',)); (x, y,) = R2._first_ngens(2)
>>> R3 = PolynomialRing(QQ, Integer(3), order='lex', names=('x', 'y', 'z',)); (x, y, z,) = R3._first_ngens(3)
>>> I2 = x*R2
>>> I3 = x*R3
>>> I2.intersection(I3)
Traceback (most recent call last):
...
TypeError: Intersection is only available for ideals of the same ring.
is_prime(**kwds)[source]

Return True if this ideal is prime.

INPUT:

EXAMPLES:

sage: R.<x, y> = PolynomialRing(QQ, 2)
sage: I = (x^2 - y^2 - 1) * R
sage: I.is_prime()
True
sage: (I^2).is_prime()
False

sage: J = (x^2 - y^2) * R
sage: J.is_prime()
False
sage: (J^3).is_prime()
False

sage: (I * J).is_prime()
False

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(2), names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> I = (x**Integer(2) - y**Integer(2) - Integer(1)) * R
>>> I.is_prime()
True
>>> (I**Integer(2)).is_prime()
False

>>> J = (x**Integer(2) - y**Integer(2)) * R
>>> J.is_prime()
False
>>> (J**Integer(3)).is_prime()
False

>>> (I * J).is_prime()
False

The following is Issue #5982. Note that the quotient ring is not recognized as being a field at this time, so the fraction field is not the quotient ring itself:

sage: Q = R.quotient(I); Q
Quotient of Multivariate Polynomial Ring in x, y over Rational Field
 by the ideal (x^2 - y^2 - 1)
sage: Q.fraction_field()
Fraction Field of
 Quotient of Multivariate Polynomial Ring in x, y over Rational Field
  by the ideal (x^2 - y^2 - 1)

>>> from sage.all import *
>>> Q = R.quotient(I); Q
Quotient of Multivariate Polynomial Ring in x, y over Rational Field
 by the ideal (x^2 - y^2 - 1)
>>> Q.fraction_field()
Fraction Field of
 Quotient of Multivariate Polynomial Ring in x, y over Rational Field
  by the ideal (x^2 - y^2 - 1)
minimal_associated_primes()[source]

OUTPUT: list of prime ideals

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(QQ, 3, 'xyz')
sage: p = z^2 + 1; q = z^3 + 2
sage: I = (p*q^2, y - z^2) * R
sage: sorted(I.minimal_associated_primes(), key=str)
[Ideal (z^2 + 1, -z^2 + y)
  of Multivariate Polynomial Ring in x, y, z over Rational Field,
 Ideal (z^3 + 2, -z^2 + y)
  of Multivariate Polynomial Ring in x, y, z over Rational Field]

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(3), 'xyz', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> p = z**Integer(2) + Integer(1); q = z**Integer(3) + Integer(2)
>>> I = (p*q**Integer(2), y - z**Integer(2)) * R
>>> sorted(I.minimal_associated_primes(), key=str)
[Ideal (z^2 + 1, -z^2 + y)
  of Multivariate Polynomial Ring in x, y, z over Rational Field,
 Ideal (z^3 + 2, -z^2 + y)
  of Multivariate Polynomial Ring in x, y, z over Rational Field]

ALGORITHM:

Uses Singular.

normal_basis(degree=None, algorithm='libsingular', singular=None)[source]

Return a vector space basis of the quotient ring of this ideal.

INPUT:

  • degree – integer (default: None)

  • algorithm – string (default: 'libsingular'); if not the default, this will use the kbase() or weightKB() command from Singular

  • singular – the singular interpreter to use when algorithm is not 'libsingular' (default: the default instance)

OUTPUT:

Monomials in the basis. If degree is given, only the monomials of the given degree are returned.

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(QQ)
sage: I = R.ideal(x^2 + y^2 + z^2 - 4, x^2 + 2*y^2 - 5, x*z - 1)
sage: I.normal_basis()
[y*z^2, z^2, y*z, z, x*y, y, x, 1]
sage: I.normal_basis(algorithm='singular')
[y*z^2, z^2, y*z, z, x*y, y, x, 1]

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> I = R.ideal(x**Integer(2) + y**Integer(2) + z**Integer(2) - Integer(4), x**Integer(2) + Integer(2)*y**Integer(2) - Integer(5), x*z - Integer(1))
>>> I.normal_basis()
[y*z^2, z^2, y*z, z, x*y, y, x, 1]
>>> I.normal_basis(algorithm='singular')
[y*z^2, z^2, y*z, z, x*y, y, x, 1]

The result can be restricted to monomials of a chosen degree, which is particularly useful when the quotient ring is not finite-dimensional as a vector space.

sage: J = R.ideal(x^2 + y^2 + z^2 - 4, x^2 + 2*y^2 - 5)
sage: J.dimension()
1
sage: [J.normal_basis(d) for d in (0..3)]
[[1], [z, y, x], [z^2, y*z, x*z, x*y], [z^3, y*z^2, x*z^2, x*y*z]]
sage: [J.normal_basis(d, algorithm='singular') for d in (0..3)]
[[1], [z, y, x], [z^2, y*z, x*z, x*y], [z^3, y*z^2, x*z^2, x*y*z]]

>>> from sage.all import *
>>> J = R.ideal(x**Integer(2) + y**Integer(2) + z**Integer(2) - Integer(4), x**Integer(2) + Integer(2)*y**Integer(2) - Integer(5))
>>> J.dimension()
1
>>> [J.normal_basis(d) for d in (ellipsis_iter(Integer(0),Ellipsis,Integer(3)))]
[[1], [z, y, x], [z^2, y*z, x*z, x*y], [z^3, y*z^2, x*z^2, x*y*z]]
>>> [J.normal_basis(d, algorithm='singular') for d in (ellipsis_iter(Integer(0),Ellipsis,Integer(3)))]
[[1], [z, y, x], [z^2, y*z, x*z, x*y], [z^3, y*z^2, x*z^2, x*y*z]]

In case of a polynomial ring with a weighted term order, the degree of the monomials is taken with respect to the weights.

sage: T = TermOrder('wdegrevlex', (1, 2, 3))
sage: R.<x,y,z> = PolynomialRing(QQ, order=T)
sage: B = R.ideal(x*y^2 + x^5, z*y + x^3*y).normal_basis(9); B
[x^2*y^2*z, x^3*z^2, x*y*z^2, z^3]
sage: all(f.degree() == 9 for f in B)
True

>>> from sage.all import *
>>> T = TermOrder('wdegrevlex', (Integer(1), Integer(2), Integer(3)))
>>> R = PolynomialRing(QQ, order=T, names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> B = R.ideal(x*y**Integer(2) + x**Integer(5), z*y + x**Integer(3)*y).normal_basis(Integer(9)); B
[x^2*y^2*z, x^3*z^2, x*y*z^2, z^3]
>>> all(f.degree() == Integer(9) for f in B)
True
plot(singular=None)[source]

If you somehow manage to install surf, perhaps you can use this function to implicitly plot the real zero locus of this ideal (if principal).

INPUT:

  • self – must be a principal ideal in 2 or 3 vars over \(\QQ\)

EXAMPLES:

Implicit plotting in 2-d:

sage: R.<x,y> = PolynomialRing(QQ,2)
sage: I = R.ideal([y^3 - x^2])
sage: I.plot()        # cusp
Graphics object consisting of 1 graphics primitive
sage: I = R.ideal([y^2 - x^2 - 1])
sage: I.plot()        # hyperbola
Graphics object consisting of 1 graphics primitive
sage: I = R.ideal([y^2 + x^2*(1/4) - 1])
sage: I.plot()        # ellipse
Graphics object consisting of 1 graphics primitive
sage: I = R.ideal([y^2-(x^2-1)*(x-2)])
sage: I.plot()        # elliptic curve
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> R = PolynomialRing(QQ,Integer(2), names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> I = R.ideal([y**Integer(3) - x**Integer(2)])
>>> I.plot()        # cusp
Graphics object consisting of 1 graphics primitive
>>> I = R.ideal([y**Integer(2) - x**Integer(2) - Integer(1)])
>>> I.plot()        # hyperbola
Graphics object consisting of 1 graphics primitive
>>> I = R.ideal([y**Integer(2) + x**Integer(2)*(Integer(1)/Integer(4)) - Integer(1)])
>>> I.plot()        # ellipse
Graphics object consisting of 1 graphics primitive
>>> I = R.ideal([y**Integer(2)-(x**Integer(2)-Integer(1))*(x-Integer(2))])
>>> I.plot()        # elliptic curve
Graphics object consisting of 1 graphics primitive

Implicit plotting in 3-d:

sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: I = R.ideal([y^2 + x^2*(1/4) - z])
sage: I.plot()          # a cone; optional - surf
sage: I = R.ideal([y^2 + z^2*(1/4) - x])
sage: I.plot()          # same code, from a different angle; optional - surf
sage: I = R.ideal([x^2*y^2+x^2*z^2+y^2*z^2-16*x*y*z])
sage: I.plot()          # Steiner surface; optional - surf

>>> from sage.all import *
>>> R = PolynomialRing(QQ,Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> I = R.ideal([y**Integer(2) + x**Integer(2)*(Integer(1)/Integer(4)) - z])
>>> I.plot()          # a cone; optional - surf
>>> I = R.ideal([y**Integer(2) + z**Integer(2)*(Integer(1)/Integer(4)) - x])
>>> I.plot()          # same code, from a different angle; optional - surf
>>> I = R.ideal([x**Integer(2)*y**Integer(2)+x**Integer(2)*z**Integer(2)+y**Integer(2)*z**Integer(2)-Integer(16)*x*y*z])
>>> I.plot()          # Steiner surface; optional - surf

AUTHORS:

  • David Joyner (2006-02-12)

primary_decomposition(algorithm='sy')[source]

Return a list of primary ideals such that their intersection is self.

An ideal \(Q\) is called primary if it is a proper ideal of the ring \(R\), and if whenever \(ab \in Q\) and \(a \not\in Q\), then \(b^n \in Q\) for some \(n \in \ZZ\).

If \(Q\) is a primary ideal of the ring \(R\), then the radical ideal \(P\) of \(Q\) (i.e., the ideal consisting of all \(a \in R\) with \(a^n \in Q\) for some \(n \in \ZZ\)), is called the associated prime of \(Q\).

If \(I\) is a proper ideal of a Noetherian ring \(R\), then there exists a finite collection of primary ideals \(Q_i\) such that the following hold:

  • the intersection of the \(Q_i\) is \(I\);

  • none of the \(Q_i\) contains the intersection of the others;

  • the associated prime ideals of the \(Q_i\) are pairwise distinct.

INPUT:

  • algorithm – string; one of

    • 'sy' – (default) use the Shimoyama-Yokoyama algorithm

    • 'gtz' – use the Gianni-Trager-Zacharias algorithm

OUTPUT:

  • a list of primary ideals \(Q_i\) forming a primary decomposition of self.

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
sage: p = z^2 + 1; q = z^3 + 2
sage: I = (p*q^2, y - z^2) * R
sage: pd = I.primary_decomposition(); sorted(pd, key=str)
[Ideal (z^2 + 1, y + 1)
  of Multivariate Polynomial Ring in x, y, z over Rational Field,
 Ideal (z^6 + 4*z^3 + 4, y - z^2)
  of Multivariate Polynomial Ring in x, y, z over Rational Field]

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(3), order='lex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> p = z**Integer(2) + Integer(1); q = z**Integer(3) + Integer(2)
>>> I = (p*q**Integer(2), y - z**Integer(2)) * R
>>> pd = I.primary_decomposition(); sorted(pd, key=str)
[Ideal (z^2 + 1, y + 1)
  of Multivariate Polynomial Ring in x, y, z over Rational Field,
 Ideal (z^6 + 4*z^3 + 4, y - z^2)
  of Multivariate Polynomial Ring in x, y, z over Rational Field]


sage: from functools import reduce
sage: reduce(lambda Qi, Qj: Qi.intersection(Qj), pd) == I
True

>>> from sage.all import *
>>> from functools import reduce
>>> reduce(lambda Qi, Qj: Qi.intersection(Qj), pd) == I
True

ALGORITHM:

Uses Singular.

REFERENCES:

  • Thomas Becker and Volker Weispfenning. Groebner Bases - A Computational Approach To Commutative Algebra. Springer, New York 1993.

primary_decomposition_complete()[source]

A decorator that creates a cached version of an instance method of a class.

Note

For proper behavior, the method must be a pure function (no side effects). Arguments to the method must be hashable or transformed into something hashable using key or they must define sage.structure.sage_object.SageObject._cache_key().

EXAMPLES:

sage: class Foo():
....:     @cached_method
....:     def f(self, t, x=2):
....:         print('computing')
....:         return t**x
sage: a = Foo()

>>> from sage.all import *
>>> class Foo():
...     @cached_method
...     def f(self, t, x=Integer(2)):
...         print('computing')
...         return t**x
>>> a = Foo()

The example shows that the actual computation takes place only once, and that the result is identical for equivalent input:

sage: res = a.f(3, 2); res
computing
9
sage: a.f(t = 3, x = 2) is res
True
sage: a.f(3) is res
True

>>> from sage.all import *
>>> res = a.f(Integer(3), Integer(2)); res
computing
9
>>> a.f(t = Integer(3), x = Integer(2)) is res
True
>>> a.f(Integer(3)) is res
True

Note, however, that the CachedMethod is replaced by a CachedMethodCaller or CachedMethodCallerNoArgs as soon as it is bound to an instance or class:

sage: P.<a,b,c,d> = QQ[]
sage: I = P*[a,b]
sage: type(I.__class__.gens)
<class 'sage.misc.cachefunc.CachedMethodCallerNoArgs'>

>>> from sage.all import *
>>> P = QQ['a, b, c, d']; (a, b, c, d,) = P._first_ngens(4)
>>> I = P*[a,b]
>>> type(I.__class__.gens)
<class 'sage.misc.cachefunc.CachedMethodCallerNoArgs'>

So, you would hardly ever see an instance of this class alive.

The parameter key can be used to pass a function which creates a custom cache key for inputs. In the following example, this parameter is used to ignore the algorithm keyword for caching:

sage: class A():
....:     def _f_normalize(self, x, algorithm): return x
....:     @cached_method(key=_f_normalize)
....:     def f(self, x, algorithm='default'): return x
sage: a = A()
sage: a.f(1, algorithm='default') is a.f(1) is a.f(1, algorithm='algorithm')
True

>>> from sage.all import *
>>> class A():
...     def _f_normalize(self, x, algorithm): return x
...     @cached_method(key=_f_normalize)
...     def f(self, x, algorithm='default'): return x
>>> a = A()
>>> a.f(Integer(1), algorithm='default') is a.f(Integer(1)) is a.f(Integer(1), algorithm='algorithm')
True

The parameter do_pickle can be used to enable pickling of the cache. Usually the cache is not stored when pickling:

sage: class A():
....:     @cached_method
....:     def f(self, x): return None
sage: import __main__
sage: __main__.A = A
sage: a = A()
sage: a.f(1)
sage: len(a.f.cache)
1
sage: b = loads(dumps(a))
sage: len(b.f.cache)
0

>>> from sage.all import *
>>> class A():
...     @cached_method
...     def f(self, x): return None
>>> import __main__
>>> __main__.A = A
>>> a = A()
>>> a.f(Integer(1))
>>> len(a.f.cache)
1
>>> b = loads(dumps(a))
>>> len(b.f.cache)
0

When do_pickle is set, the pickle contains the contents of the cache:

sage: class A():
....:     @cached_method(do_pickle=True)
....:     def f(self, x): return None
sage: __main__.A = A
sage: a = A()
sage: a.f(1)
sage: len(a.f.cache)
1
sage: b = loads(dumps(a))
sage: len(b.f.cache)
1

>>> from sage.all import *
>>> class A():
...     @cached_method(do_pickle=True)
...     def f(self, x): return None
>>> __main__.A = A
>>> a = A()
>>> a.f(Integer(1))
>>> len(a.f.cache)
1
>>> b = loads(dumps(a))
>>> len(b.f.cache)
1

Cached methods cannot be copied like usual methods, see Issue #12603. Copying them can lead to very surprising results:

sage: class A:
....:     @cached_method
....:     def f(self):
....:         return 1
sage: class B:
....:     g=A.f
....:     def f(self):
....:         return 2

sage: b=B()
sage: b.f()
2
sage: b.g()
1
sage: b.f()
1

>>> from sage.all import *
>>> class A:
...     @cached_method
...     def f(self):
...         return Integer(1)
>>> class B:
...     g=A.f
...     def f(self):
...         return Integer(2)

>>> b=B()
>>> b.f()
2
>>> b.g()
1
>>> b.f()
1
quotient(J)[source]

Given ideals \(I\) = self and \(J\) in the same polynomial ring \(P\), return the ideal quotient of \(I\) by \(J\) consisting of the polynomials \(a\) of \(P\) such that \(\{aJ \subset I\}\).

This is also referred to as the colon ideal (\(I\):\(J\)).

INPUT:

  • J – multivariate polynomial ideal

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: R.<x,y,z> = PolynomialRing(GF(181), 3)
sage: I = Ideal([x^2 + x*y*z, y^2 - z^3*y, z^3 + y^5*x*z])
sage: J = Ideal([x])
sage: Q = I.quotient(J)
sage: y*z + x in I
False
sage: x in J
True
sage: x * (y*z + x) in I
True

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> R = PolynomialRing(GF(Integer(181)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> I = Ideal([x**Integer(2) + x*y*z, y**Integer(2) - z**Integer(3)*y, z**Integer(3) + y**Integer(5)*x*z])
>>> J = Ideal([x])
>>> Q = I.quotient(J)
>>> y*z + x in I
False
>>> x in J
True
>>> x * (y*z + x) in I
True
radical()[source]

The radical of this ideal.

EXAMPLES:

This is an obviously not radical ideal:

sage: R.<x,y,z> = PolynomialRing(QQ, 3)
sage: I = (x^2, y^3, (x*z)^4 + y^3 + 10*x^2) * R
sage: I.radical()
Ideal (y, x) of Multivariate Polynomial Ring in x, y, z over Rational Field

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> I = (x**Integer(2), y**Integer(3), (x*z)**Integer(4) + y**Integer(3) + Integer(10)*x**Integer(2)) * R
>>> I.radical()
Ideal (y, x) of Multivariate Polynomial Ring in x, y, z over Rational Field

That the radical is correct is clear from the Groebner basis.

sage: I.groebner_basis()
[y^3, x^2]

>>> from sage.all import *
>>> I.groebner_basis()
[y^3, x^2]

This is the example from the Singular manual:

sage: p = z^2 + 1; q = z^3 + 2
sage: I = (p*q^2, y - z^2) * R
sage: I.radical()
Ideal (z^2 - y, y^2*z + y*z + 2*y + 2)
 of Multivariate Polynomial Ring in x, y, z over Rational Field

>>> from sage.all import *
>>> p = z**Integer(2) + Integer(1); q = z**Integer(3) + Integer(2)
>>> I = (p*q**Integer(2), y - z**Integer(2)) * R
>>> I.radical()
Ideal (z^2 - y, y^2*z + y*z + 2*y + 2)
 of Multivariate Polynomial Ring in x, y, z over Rational Field

Note

From the Singular manual: A combination of the algorithms of Krick/Logar and Kemper is used. Works also in positive characteristic (Kempers algorithm).

sage: # needs sage.rings.finite_rings
sage: R.<x,y,z> = PolynomialRing(GF(37), 3)
sage: p = z^2 + 1; q = z^3 + 2
sage: I = (p*q^2, y - z^2) * R
sage: I.radical()
Ideal (z^2 - y, y^2*z + y*z + 2*y + 2)
 of Multivariate Polynomial Ring in x, y, z over Finite Field of size 37

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> R = PolynomialRing(GF(Integer(37)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> p = z**Integer(2) + Integer(1); q = z**Integer(3) + Integer(2)
>>> I = (p*q**Integer(2), y - z**Integer(2)) * R
>>> I.radical()
Ideal (z^2 - y, y^2*z + y*z + 2*y + 2)
 of Multivariate Polynomial Ring in x, y, z over Finite Field of size 37
saturation(other)[source]

Return the saturation (and saturation exponent) of the ideal self with respect to the ideal other.

INPUT:

  • other – another ideal in the same ring

OUTPUT: a pair (ideal, integer)

EXAMPLES:

sage: R.<x, y, z> = QQ[]
sage: I = R.ideal(x^5*z^3, x*y*z, y*z^4)
sage: J = R.ideal(z)
sage: I.saturation(J)
(Ideal (y, x^5) of Multivariate Polynomial Ring in x, y, z over Rational Field, 4)

>>> from sage.all import *
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> I = R.ideal(x**Integer(5)*z**Integer(3), x*y*z, y*z**Integer(4))
>>> J = R.ideal(z)
>>> I.saturation(J)
(Ideal (y, x^5) of Multivariate Polynomial Ring in x, y, z over Rational Field, 4)
syzygy_module()[source]

Compute the first syzygy (i.e., the module of relations of the given generators) of the ideal.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ)
sage: f = 2*x^2 + y
sage: g = y
sage: h = 2*f + g
sage: I = Ideal([f,g,h])
sage: M = I.syzygy_module(); M
[       -2        -1         1]
[       -y 2*x^2 + y         0]
sage: G = vector(I.gens())
sage: M*G
(0, 0)

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(2)*x**Integer(2) + y
>>> g = y
>>> h = Integer(2)*f + g
>>> I = Ideal([f,g,h])
>>> M = I.syzygy_module(); M
[       -2        -1         1]
[       -y 2*x^2 + y         0]
>>> G = vector(I.gens())
>>> M*G
(0, 0)

ALGORITHM:

Uses Singular’s syz command.

transformed_basis(algorithm='gwalk', other_ring=None, singular=None)[source]

Return a lex or other_ring Groebner Basis for this ideal.

INPUT:

  • algorithm – see below for options

  • other_ring – only valid for algorithm='fglm'; if provided, conversion will be performed to this ring. Otherwise a lex Groebner basis will be returned.

ALGORITHMS:

  • 'fglm' – FGLM algorithm. The input ideal must be given with a reduced Groebner Basis of a zero-dimensional ideal

  • 'gwalk' – Groebner Walk algorithm (default)

  • 'awalk1' – ‘first alternative’ algorithm

  • 'awalk2' – ‘second alternative’ algorithm

  • 'twalk' – Tran algorithm

  • 'fwalk' – Fractal Walk algorithm

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: I = Ideal([y^3+x^2,x^2*y+x^2, x^3-x^2, z^4-x^2-y])
sage: I = Ideal(I.groebner_basis())
sage: S.<z,x,y> = PolynomialRing(QQ,3,order='lex')
sage: J = Ideal(I.transformed_basis('fglm',S))
sage: J
Ideal (z^4 + y^3 - y, x^2 + y^3, x*y^3 - y^3, y^4 + y^3)
 of Multivariate Polynomial Ring in z, x, y over Rational Field

>>> from sage.all import *
>>> R = PolynomialRing(QQ,Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> I = Ideal([y**Integer(3)+x**Integer(2),x**Integer(2)*y+x**Integer(2), x**Integer(3)-x**Integer(2), z**Integer(4)-x**Integer(2)-y])
>>> I = Ideal(I.groebner_basis())
>>> S = PolynomialRing(QQ,Integer(3),order='lex', names=('z', 'x', 'y',)); (z, x, y,) = S._first_ngens(3)
>>> J = Ideal(I.transformed_basis('fglm',S))
>>> J
Ideal (z^4 + y^3 - y, x^2 + y^3, x*y^3 - y^3, y^4 + y^3)
 of Multivariate Polynomial Ring in z, x, y over Rational Field


sage: R.<z,y,x> = PolynomialRing(GF(32003), 3, order='lex')                 # needs sage.rings.finite_rings
sage: I = Ideal([y^3 + x*y*z + y^2*z + x*z^3, 3 + x*y + x^2*y + y^2*z])     # needs sage.rings.finite_rings
sage: I.transformed_basis('gwalk')                                          # needs sage.rings.finite_rings
[z*y^2 + y*x^2 + y*x + 3,
 z*x + 8297*y^8*x^2 + 8297*y^8*x + 3556*y^7 - 8297*y^6*x^4 + 15409*y^6*x^3
   - 8297*y^6*x^2 - 8297*y^5*x^5 + 15409*y^5*x^4 - 8297*y^5*x^3 + 3556*y^5*x^2
   + 3556*y^5*x + 3556*y^4*x^3 + 3556*y^4*x^2 - 10668*y^4 - 10668*y^3*x
   - 8297*y^2*x^9 - 1185*y^2*x^8 + 14224*y^2*x^7 - 1185*y^2*x^6 - 8297*y^2*x^5
   - 14223*y*x^7 - 10666*y*x^6 - 10666*y*x^5 - 14223*y*x^4 + x^5 + 2*x^4 + x^3,
 y^9 - y^7*x^2 - y^7*x - y^6*x^3 - y^6*x^2 - 3*y^6 - 3*y^5*x - y^3*x^7
   - 3*y^3*x^6 - 3*y^3*x^5 - y^3*x^4 - 9*y^2*x^5 - 18*y^2*x^4 - 9*y^2*x^3
   - 27*y*x^3 - 27*y*x^2 - 27*x]

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(32003)), Integer(3), order='lex', names=('z', 'y', 'x',)); (z, y, x,) = R._first_ngens(3)# needs sage.rings.finite_rings
>>> I = Ideal([y**Integer(3) + x*y*z + y**Integer(2)*z + x*z**Integer(3), Integer(3) + x*y + x**Integer(2)*y + y**Integer(2)*z])     # needs sage.rings.finite_rings
>>> I.transformed_basis('gwalk')                                          # needs sage.rings.finite_rings
[z*y^2 + y*x^2 + y*x + 3,
 z*x + 8297*y^8*x^2 + 8297*y^8*x + 3556*y^7 - 8297*y^6*x^4 + 15409*y^6*x^3
   - 8297*y^6*x^2 - 8297*y^5*x^5 + 15409*y^5*x^4 - 8297*y^5*x^3 + 3556*y^5*x^2
   + 3556*y^5*x + 3556*y^4*x^3 + 3556*y^4*x^2 - 10668*y^4 - 10668*y^3*x
   - 8297*y^2*x^9 - 1185*y^2*x^8 + 14224*y^2*x^7 - 1185*y^2*x^6 - 8297*y^2*x^5
   - 14223*y*x^7 - 10666*y*x^6 - 10666*y*x^5 - 14223*y*x^4 + x^5 + 2*x^4 + x^3,
 y^9 - y^7*x^2 - y^7*x - y^6*x^3 - y^6*x^2 - 3*y^6 - 3*y^5*x - y^3*x^7
   - 3*y^3*x^6 - 3*y^3*x^5 - y^3*x^4 - 9*y^2*x^5 - 18*y^2*x^4 - 9*y^2*x^3
   - 27*y*x^3 - 27*y*x^2 - 27*x]

ALGORITHM:

Uses Singular.

triangular_decomposition(algorithm=None, singular=None)[source]

Decompose zero-dimensional ideal self into triangular sets.

This requires that the given basis is reduced w.r.t. to the lexicographical monomial ordering. If the basis of self does not have this property, the required Groebner basis is computed implicitly.

INPUT:

  • algorithm – string or None (default: None)

ALGORITHMS:

  • 'singular:triangL' – decomposition of self into triangular systems (Lazard)

  • 'singular:triangLfak' – decomposition of self into triangular systems plus factorization

  • 'singular:triangM' – decomposition of self into triangular systems (Moeller)

OUTPUT: list \(T\) of lists \(t\) such that the variety of self is the union of the varieties of \(t\) in \(L\) and each \(t\) is in triangular form

EXAMPLES:

sage: P.<e,d,c,b,a> = PolynomialRing(QQ, 5, order='lex')
sage: I = sage.rings.ideal.Cyclic(P)
sage: GB = Ideal(I.groebner_basis('libsingular:stdfglm'))
sage: GB.triangular_decomposition('singular:triangLfak')
[Ideal (a - 1, b - 1, c - 1, d^2 + 3*d + 1, e + d + 3) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a - 1, b - 1, c^2 + 3*c + 1, d + c + 3, e - 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a - 1, b^2 + 3*b + 1, c + b + 3, d - 1, e - 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a - 1, b^4 + b^3 + b^2 + b + 1, -c + b^2, -d + b^3,
        e + b^3 + b^2 + b + 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^2 + 3*a + 1, b - 1, c - 1, d - 1, e + a + 3) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^2 + 3*a + 1, b + a + 3, c - 1, d - 1, e - 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^4 - 4*a^3 + 6*a^2 + a + 1,
        -11*b^2 + 6*b*a^3 - 26*b*a^2 + 41*b*a - 4*b - 8*a^3 + 31*a^2 - 40*a - 24,
        11*c + 3*a^3 - 13*a^2 + 26*a - 2, 11*d + 3*a^3 - 13*a^2 + 26*a - 2,
        -11*e - 11*b + 6*a^3 - 26*a^2 + 41*a - 4) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^4 + a^3 + a^2 + a + 1,
        b - 1, c + a^3 + a^2 + a + 1, -d + a^3, -e + a^2) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^4 + a^3 + a^2 + a + 1,
        b - a, c - a, d^2 + 3*d*a + a^2, e + d + 3*a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^4 + a^3 + a^2 + a + 1,
        b - a, c^2 + 3*c*a + a^2, d + c + 3*a, e - a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^4 + a^3 + a^2 + a + 1,
        b^2 + 3*b*a + a^2, c + b + 3*a, d - a, e - a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^4 + a^3 + a^2 + a + 1,
        b^3 + b^2*a + b^2 + b*a^2 + b*a + b + a^3 + a^2 + a + 1,
        c + b^2*a^3 + b^2*a^2 + b^2*a + b^2,
        -d + b^2*a^2 + b^2*a + b^2 + b*a^2 + b*a + a^2,
        -e + b^2*a^3 - b*a^2 - b*a - b - a^2 - a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^4 + a^3 + 6*a^2 - 4*a + 1,
        -11*b^2 + 6*b*a^3 + 10*b*a^2 + 39*b*a + 2*b + 16*a^3 + 23*a^2 + 104*a - 24,
        11*c + 3*a^3 + 5*a^2 + 25*a + 1, 11*d + 3*a^3 + 5*a^2 + 25*a + 1,
        -11*e - 11*b + 6*a^3 + 10*a^2 + 39*a + 2) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field]

sage: R.<x1,x2> = PolynomialRing(QQ, 2, order='lex')
sage: f1 = 1/2*((x1^2 + 2*x1 - 4)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2)
sage: f2 = 1/2*((x1^2 + 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 - 4*x1^2)
sage: I = Ideal(f1,f2)
sage: I.triangular_decomposition()
[Ideal (x2, x1^2) of Multivariate Polynomial Ring in x1, x2 over Rational Field,
 Ideal (x2, x1^2) of Multivariate Polynomial Ring in x1, x2 over Rational Field,
 Ideal (x2, x1^2) of Multivariate Polynomial Ring in x1, x2 over Rational Field,
 Ideal (x2^4 + 4*x2^3 - 6*x2^2 - 20*x2 + 5, 8*x1 - x2^3 + x2^2 + 13*x2 - 5)
  of Multivariate Polynomial Ring in x1, x2 over Rational Field]

>>> from sage.all import *
>>> P = PolynomialRing(QQ, Integer(5), order='lex', names=('e', 'd', 'c', 'b', 'a',)); (e, d, c, b, a,) = P._first_ngens(5)
>>> I = sage.rings.ideal.Cyclic(P)
>>> GB = Ideal(I.groebner_basis('libsingular:stdfglm'))
>>> GB.triangular_decomposition('singular:triangLfak')
[Ideal (a - 1, b - 1, c - 1, d^2 + 3*d + 1, e + d + 3) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a - 1, b - 1, c^2 + 3*c + 1, d + c + 3, e - 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a - 1, b^2 + 3*b + 1, c + b + 3, d - 1, e - 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a - 1, b^4 + b^3 + b^2 + b + 1, -c + b^2, -d + b^3,
        e + b^3 + b^2 + b + 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^2 + 3*a + 1, b - 1, c - 1, d - 1, e + a + 3) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^2 + 3*a + 1, b + a + 3, c - 1, d - 1, e - 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^4 - 4*a^3 + 6*a^2 + a + 1,
        -11*b^2 + 6*b*a^3 - 26*b*a^2 + 41*b*a - 4*b - 8*a^3 + 31*a^2 - 40*a - 24,
        11*c + 3*a^3 - 13*a^2 + 26*a - 2, 11*d + 3*a^3 - 13*a^2 + 26*a - 2,
        -11*e - 11*b + 6*a^3 - 26*a^2 + 41*a - 4) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^4 + a^3 + a^2 + a + 1,
        b - 1, c + a^3 + a^2 + a + 1, -d + a^3, -e + a^2) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^4 + a^3 + a^2 + a + 1,
        b - a, c - a, d^2 + 3*d*a + a^2, e + d + 3*a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^4 + a^3 + a^2 + a + 1,
        b - a, c^2 + 3*c*a + a^2, d + c + 3*a, e - a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^4 + a^3 + a^2 + a + 1,
        b^2 + 3*b*a + a^2, c + b + 3*a, d - a, e - a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^4 + a^3 + a^2 + a + 1,
        b^3 + b^2*a + b^2 + b*a^2 + b*a + b + a^3 + a^2 + a + 1,
        c + b^2*a^3 + b^2*a^2 + b^2*a + b^2,
        -d + b^2*a^2 + b^2*a + b^2 + b*a^2 + b*a + a^2,
        -e + b^2*a^3 - b*a^2 - b*a - b - a^2 - a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
 Ideal (a^4 + a^3 + 6*a^2 - 4*a + 1,
        -11*b^2 + 6*b*a^3 + 10*b*a^2 + 39*b*a + 2*b + 16*a^3 + 23*a^2 + 104*a - 24,
        11*c + 3*a^3 + 5*a^2 + 25*a + 1, 11*d + 3*a^3 + 5*a^2 + 25*a + 1,
        -11*e - 11*b + 6*a^3 + 10*a^2 + 39*a + 2) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field]

>>> R = PolynomialRing(QQ, Integer(2), order='lex', names=('x1', 'x2',)); (x1, x2,) = R._first_ngens(2)
>>> f1 = Integer(1)/Integer(2)*((x1**Integer(2) + Integer(2)*x1 - Integer(4))*x2**Integer(2) + Integer(2)*(x1**Integer(2) + x1)*x2 + x1**Integer(2))
>>> f2 = Integer(1)/Integer(2)*((x1**Integer(2) + Integer(2)*x1 + Integer(1))*x2**Integer(2) + Integer(2)*(x1**Integer(2) + x1)*x2 - Integer(4)*x1**Integer(2))
>>> I = Ideal(f1,f2)
>>> I.triangular_decomposition()
[Ideal (x2, x1^2) of Multivariate Polynomial Ring in x1, x2 over Rational Field,
 Ideal (x2, x1^2) of Multivariate Polynomial Ring in x1, x2 over Rational Field,
 Ideal (x2, x1^2) of Multivariate Polynomial Ring in x1, x2 over Rational Field,
 Ideal (x2^4 + 4*x2^3 - 6*x2^2 - 20*x2 + 5, 8*x1 - x2^3 + x2^2 + 13*x2 - 5)
  of Multivariate Polynomial Ring in x1, x2 over Rational Field]
variety(ring=None)[source]

Return the variety of this ideal.

Given a zero-dimensional ideal \(I\) (= self) of a polynomial ring \(P\) whose order is lexicographic, return the variety of \(I\) as a list of dictionaries with (variable, value) pairs. By default, the variety of the ideal over its coefficient field \(K\) is returned; ring can be specified to find the variety over a different ring.

These dictionaries have cardinality equal to the number of variables in \(P\) and represent assignments of values to these variables such that all polynomials in \(I\) vanish.

If ring is specified, then a triangular decomposition of self is found over the original coefficient field \(K\); then the triangular systems are solved using root-finding over ring. This is particularly useful when \(K\) is QQ (to allow fast symbolic computation of the triangular decomposition) and ring is RR, AA, CC, or QQbar (to compute the whole real or complex variety of the ideal).

Note that with ring=RR or CC, computation is done numerically and potentially inaccurately; in particular, the number of points in the real variety may be miscomputed. With ring=AA or QQbar, computation is done exactly (which may be much slower, of course).

INPUT:

  • ring – return roots in the ring instead of the base ring of this ideal (default: None)

  • algorithm – algorithm or implementation to use; see below for supported values

  • proof – return a provably correct result (default: True)

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<w> = GF(27)  # this example is from the MAGMA handbook
sage: P.<x, y> = PolynomialRing(K, 2, order='lex')
sage: I = Ideal([x^8 + y + 2, y^6 + x*y^5 + x^2])
sage: I = Ideal(I.groebner_basis()); I
Ideal (x - y^47 - y^45 + y^44 - y^43 + y^41 - y^39 - y^38 - y^37 - y^36
         + y^35 - y^34 - y^33 + y^32 - y^31 + y^30 + y^28 + y^27 + y^26
         + y^25 - y^23 + y^22 + y^21 - y^19 - y^18 - y^16 + y^15 + y^13
         + y^12 - y^10 + y^9 + y^8 + y^7 - y^6 + y^4 + y^3 + y^2 + y - 1,
       y^48 + y^41 - y^40 + y^37 - y^36 - y^33 + y^32 - y^29 + y^28
         - y^25 + y^24 + y^2 + y + 1)
of Multivariate Polynomial Ring in x, y over Finite Field in w of size 3^3
sage: V = I.variety();
sage: sorted(V, key=str)
[{y: w^2 + 2*w, x: 2*w + 2}, {y: w^2 + 2, x: 2*w}, {y: w^2 + w, x: 2*w + 1}]
sage: [f.subs(v)                      # check that all polynomials vanish
....:  for f in I.gens() for v in V]
[0, 0, 0, 0, 0, 0]
sage: [I.subs(v).is_zero() for v in V]  # same test, but nicer syntax
[True, True, True]

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> K = GF(Integer(27), names=('w',)); (w,) = K._first_ngens(1)# this example is from the MAGMA handbook
>>> P = PolynomialRing(K, Integer(2), order='lex', names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> I = Ideal([x**Integer(8) + y + Integer(2), y**Integer(6) + x*y**Integer(5) + x**Integer(2)])
>>> I = Ideal(I.groebner_basis()); I
Ideal (x - y^47 - y^45 + y^44 - y^43 + y^41 - y^39 - y^38 - y^37 - y^36
         + y^35 - y^34 - y^33 + y^32 - y^31 + y^30 + y^28 + y^27 + y^26
         + y^25 - y^23 + y^22 + y^21 - y^19 - y^18 - y^16 + y^15 + y^13
         + y^12 - y^10 + y^9 + y^8 + y^7 - y^6 + y^4 + y^3 + y^2 + y - 1,
       y^48 + y^41 - y^40 + y^37 - y^36 - y^33 + y^32 - y^29 + y^28
         - y^25 + y^24 + y^2 + y + 1)
of Multivariate Polynomial Ring in x, y over Finite Field in w of size 3^3
>>> V = I.variety();
>>> sorted(V, key=str)
[{y: w^2 + 2*w, x: 2*w + 2}, {y: w^2 + 2, x: 2*w}, {y: w^2 + w, x: 2*w + 1}]
>>> [f.subs(v)                      # check that all polynomials vanish
...  for f in I.gens() for v in V]
[0, 0, 0, 0, 0, 0]
>>> [I.subs(v).is_zero() for v in V]  # same test, but nicer syntax
[True, True, True]

However, we only account for solutions in the ground field and not in the algebraic closure:

sage: I.vector_space_dimension()                                            # needs sage.rings.finite_rings
48

>>> from sage.all import *
>>> I.vector_space_dimension()                                            # needs sage.rings.finite_rings
48

Here we compute the points of intersection of a hyperbola and a circle, in several fields:

sage: K.<x, y> = PolynomialRing(QQ, 2, order='lex')
sage: I = Ideal([x*y - 1, (x-2)^2 + (y-1)^2 - 1])
sage: I = Ideal(I.groebner_basis()); I
Ideal (x + y^3 - 2*y^2 + 4*y - 4, y^4 - 2*y^3 + 4*y^2 - 4*y + 1)
 of Multivariate Polynomial Ring in x, y over Rational Field

>>> from sage.all import *
>>> K = PolynomialRing(QQ, Integer(2), order='lex', names=('x', 'y',)); (x, y,) = K._first_ngens(2)
>>> I = Ideal([x*y - Integer(1), (x-Integer(2))**Integer(2) + (y-Integer(1))**Integer(2) - Integer(1)])
>>> I = Ideal(I.groebner_basis()); I
Ideal (x + y^3 - 2*y^2 + 4*y - 4, y^4 - 2*y^3 + 4*y^2 - 4*y + 1)
 of Multivariate Polynomial Ring in x, y over Rational Field

These two curves have one rational intersection:

sage: I.variety()
[{y: 1, x: 1}]

>>> from sage.all import *
>>> I.variety()
[{y: 1, x: 1}]

There are two real intersections:

sage: sorted(I.variety(ring=RR), key=str)
[{y: 0.361103080528647, x: 2.76929235423863},
 {y: 1.00000000000000, x: 1.00000000000000}]
sage: I.variety(ring=AA)                                                    # needs sage.rings.number_field
[{y: 1, x: 1},
 {y: 0.3611030805286474?, x: 2.769292354238632?}]

>>> from sage.all import *
>>> sorted(I.variety(ring=RR), key=str)
[{y: 0.361103080528647, x: 2.76929235423863},
 {y: 1.00000000000000, x: 1.00000000000000}]
>>> I.variety(ring=AA)                                                    # needs sage.rings.number_field
[{y: 1, x: 1},
 {y: 0.3611030805286474?, x: 2.769292354238632?}]

and a total of four intersections:

sage: sorted(I.variety(ring=CC), key=str)
[{y: 0.31944845973567... + 1.6331702409152...*I,
  x: 0.11535382288068... - 0.58974280502220...*I},
 {y: 0.31944845973567... - 1.6331702409152...*I,
  x: 0.11535382288068... + 0.58974280502220...*I},
 {y: 0.36110308052864..., x: 2.7692923542386...},
 {y: 1.00000000000000, x: 1.00000000000000}]
sage: sorted(I.variety(ring=QQbar), key=str)                                # needs sage.rings.number_field
[{y: 0.3194484597356763? + 1.633170240915238?*I,
  x: 0.11535382288068429? - 0.5897428050222055?*I},
 {y: 0.3194484597356763? - 1.633170240915238?*I,
  x: 0.11535382288068429? + 0.5897428050222055?*I},
 {y: 0.3611030805286474?, x: 2.769292354238632?},
 {y: 1, x: 1}]

>>> from sage.all import *
>>> sorted(I.variety(ring=CC), key=str)
[{y: 0.31944845973567... + 1.6331702409152...*I,
  x: 0.11535382288068... - 0.58974280502220...*I},
 {y: 0.31944845973567... - 1.6331702409152...*I,
  x: 0.11535382288068... + 0.58974280502220...*I},
 {y: 0.36110308052864..., x: 2.7692923542386...},
 {y: 1.00000000000000, x: 1.00000000000000}]
>>> sorted(I.variety(ring=QQbar), key=str)                                # needs sage.rings.number_field
[{y: 0.3194484597356763? + 1.633170240915238?*I,
  x: 0.11535382288068429? - 0.5897428050222055?*I},
 {y: 0.3194484597356763? - 1.633170240915238?*I,
  x: 0.11535382288068429? + 0.5897428050222055?*I},
 {y: 0.3611030805286474?, x: 2.769292354238632?},
 {y: 1, x: 1}]

We can also use the optional package msolve to compute the variety. See msolve for more information.

sage: I.variety(RBF, algorithm='msolve', proof=False)   # optional - msolve
[{x: [2.76929235423863 +/- 2.08e-15], y: [0.361103080528647 +/- 4.53e-16]},
 {x: 1.000000000000000, y: 1.000000000000000}]

>>> from sage.all import *
>>> I.variety(RBF, algorithm='msolve', proof=False)   # optional - msolve
[{x: [2.76929235423863 +/- 2.08e-15], y: [0.361103080528647 +/- 4.53e-16]},
 {x: 1.000000000000000, y: 1.000000000000000}]

Computation over floating point numbers may compute only a partial solution, or even none at all. Notice that x values are missing from the following variety:

sage: R.<x,y> = CC[]
sage: I = ideal([x^2+y^2-1,x*y-1])
sage: sorted(I.variety(), key=str)
verbose 0 (...: multi_polynomial_ideal.py, variety) Warning: computations in the complex field are inexact; variety may be computed partially or incorrectly.
verbose 0 (...: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.
[{y: -0.86602540378443... + 0.500000000000000*I},
 {y: -0.86602540378443... - 0.500000000000000*I},
 {y: 0.86602540378443... + 0.500000000000000*I},
 {y: 0.86602540378443... - 0.500000000000000*I}]

>>> from sage.all import *
>>> R = CC['x, y']; (x, y,) = R._first_ngens(2)
>>> I = ideal([x**Integer(2)+y**Integer(2)-Integer(1),x*y-Integer(1)])
>>> sorted(I.variety(), key=str)
verbose 0 (...: multi_polynomial_ideal.py, variety) Warning: computations in the complex field are inexact; variety may be computed partially or incorrectly.
verbose 0 (...: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.
[{y: -0.86602540378443... + 0.500000000000000*I},
 {y: -0.86602540378443... - 0.500000000000000*I},
 {y: 0.86602540378443... + 0.500000000000000*I},
 {y: 0.86602540378443... - 0.500000000000000*I}]

This is due to precision error, which causes the computation of an intermediate Groebner basis to fail.

If the ground field’s characteristic is too large for Singular, we resort to a toy implementation:

sage: # needs sage.rings.finite_rings
sage: R.<x,y> = PolynomialRing(GF(2147483659^3), order='lex')
sage: I = ideal([x^3 - 2*y^2, 3*x + y^4])
sage: I.variety()
verbose 0 (...: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
verbose 0 (...: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.
[{y: 0, x: 0}]

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> R = PolynomialRing(GF(Integer(2147483659)**Integer(3)), order='lex', names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> I = ideal([x**Integer(3) - Integer(2)*y**Integer(2), Integer(3)*x + y**Integer(4)])
>>> I.variety()
verbose 0 (...: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
verbose 0 (...: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.
[{y: 0, x: 0}]

The dictionary expressing the variety will be indexed by generators of the polynomial ring after changing to the target field. But the mapping will also accept generators of the original ring, or even generator names as strings, when provided as keys:

sage: # needs sage.rings.number_field
sage: K.<x,y> = QQ[]
sage: I = ideal([x^2 + 2*y - 5, x + y + 3])
sage: v = I.variety(AA)[0]; v[x], v[y]
(4.464101615137755?, -7.464101615137755?)
sage: list(v)[0].parent()
Multivariate Polynomial Ring in x, y over Algebraic Real Field
sage: v[x]
4.464101615137755?
sage: v["y"]
-7.464101615137755?

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = QQ['x, y']; (x, y,) = K._first_ngens(2)
>>> I = ideal([x**Integer(2) + Integer(2)*y - Integer(5), x + y + Integer(3)])
>>> v = I.variety(AA)[Integer(0)]; v[x], v[y]
(4.464101615137755?, -7.464101615137755?)
>>> list(v)[Integer(0)].parent()
Multivariate Polynomial Ring in x, y over Algebraic Real Field
>>> v[x]
4.464101615137755?
>>> v["y"]
-7.464101615137755?

msolve also works over finite fields:

sage: R.<x, y> = PolynomialRing(GF(536870909), 2, order='lex')              # needs sage.rings.finite_rings
sage: I = Ideal([x^2 - 1, y^2 - 1])                                         # needs sage.rings.finite_rings
sage: sorted(I.variety(algorithm='msolve',          # optional - msolve, needs sage.rings.finite_rings
....:                  proof=False),
....:        key=str)
[{x: 1, y: 1},
 {x: 1, y: 536870908},
 {x: 536870908, y: 1},
 {x: 536870908, y: 536870908}]

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(536870909)), Integer(2), order='lex', names=('x', 'y',)); (x, y,) = R._first_ngens(2)# needs sage.rings.finite_rings
>>> I = Ideal([x**Integer(2) - Integer(1), y**Integer(2) - Integer(1)])                                         # needs sage.rings.finite_rings
>>> sorted(I.variety(algorithm='msolve',          # optional - msolve, needs sage.rings.finite_rings
...                  proof=False),
...        key=str)
[{x: 1, y: 1},
 {x: 1, y: 536870908},
 {x: 536870908, y: 1},
 {x: 536870908, y: 536870908}]

but may fail in small characteristic, especially with ideals of high degree with respect to the characteristic:

sage: R.<x, y> = PolynomialRing(GF(3), 2, order='lex')
sage: I = Ideal([x^2 - 1, y^2 - 1])
sage: I.variety(algorithm='msolve', proof=False)    # optional - msolve
Traceback (most recent call last):
...
NotImplementedError: characteristic 3 too small

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(3)), Integer(2), order='lex', names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> I = Ideal([x**Integer(2) - Integer(1), y**Integer(2) - Integer(1)])
>>> I.variety(algorithm='msolve', proof=False)    # optional - msolve
Traceback (most recent call last):
...
NotImplementedError: characteristic 3 too small

ALGORITHM:

  • With algorithm = 'triangular_decomposition' (default), uses triangular decomposition, via Singular if possible, falling back on a toy implementation otherwise.

  • With algorithm = 'msolve', uses the optional package msolve. Note that msolve uses heuristics and therefore requires setting the proof flag to False. See msolve for more information.

vector_space_dimension()[source]

Return the vector space dimension of the ring modulo this ideal. If the ideal is not zero-dimensional, a TypeError is raised.

ALGORITHM:

Uses Singular.

EXAMPLES:

sage: R.<u,v> = PolynomialRing(QQ)
sage: g = u^4 + v^4 + u^3 + v^3
sage: I = ideal(g) + ideal(g.gradient())
sage: I.dimension()
0
sage: I.vector_space_dimension()
4

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('u', 'v',)); (u, v,) = R._first_ngens(2)
>>> g = u**Integer(4) + v**Integer(4) + u**Integer(3) + v**Integer(3)
>>> I = ideal(g) + ideal(g.gradient())
>>> I.dimension()
0
>>> I.vector_space_dimension()
4

When the ideal is not zero-dimensional, we return infinity:

sage: R.<x,y> = PolynomialRing(QQ)
sage: I = R.ideal(x)
sage: I.dimension()
1
sage: I.vector_space_dimension()
+Infinity

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> I = R.ideal(x)
>>> I.dimension()
1
>>> I.vector_space_dimension()
+Infinity

Due to integer overflow, the result is correct only modulo 2^32, see Issue #8586:

sage: P.<x,y,z> = PolynomialRing(GF(32003), 3)                              # needs sage.rings.finite_rings
sage: sage.rings.ideal.FieldIdeal(P).vector_space_dimension()       # known bug, needs sage.rings.finite_rings
32777216864027

>>> from sage.all import *
>>> P = PolynomialRing(GF(Integer(32003)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)# needs sage.rings.finite_rings
>>> sage.rings.ideal.FieldIdeal(P).vector_space_dimension()       # known bug, needs sage.rings.finite_rings
32777216864027
class sage.rings.polynomial.multi_polynomial_ideal.NCPolynomialIdeal(ring, gens, coerce=True, side='left')[source]

Bases: MPolynomialIdeal_singular_repr, Ideal_nc

Create a non-commutative polynomial ideal.

INPUT:

  • ring – the g-algebra to which this ideal belongs

  • gens – the generators of this ideal

  • coerce – boolean (default: True); generators are coerced into the ring before creating the ideal

  • side – (optional) string; either 'left' (default) or 'twosided'. Defines whether this ideal is left or two-sided.

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H = A.g_algebra({y*x: x*y-z, z*x: x*z+2*x, z*y: y*z-2*y})
sage: H.inject_variables()
Defining x, y, z
sage: I = H.ideal([y^2, x^2, z^2 - H.one()],             # indirect doctest
....:             coerce=False)
sage: I  # random
Left Ideal (y^2, x^2, z^2 - 1) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
sage: sorted(I.gens(), key=str)
[x^2, y^2, z^2 - 1]
sage: H.ideal([y^2, x^2, z^2 - H.one()], side='twosided')  # random
Twosided Ideal (y^2, x^2, z^2 - 1) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
sage: sorted(H.ideal([y^2, x^2, z^2 - H.one()], side='twosided').gens(),
....:        key=str)
[x^2, y^2, z^2 - 1]
sage: H.ideal([y^2, x^2, z^2 - H.one()], side='right')
Traceback (most recent call last):
...
ValueError: Only left and two-sided ideals are allowed.

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> H = A.g_algebra({y*x: x*y-z, z*x: x*z+Integer(2)*x, z*y: y*z-Integer(2)*y})
>>> H.inject_variables()
Defining x, y, z
>>> I = H.ideal([y**Integer(2), x**Integer(2), z**Integer(2) - H.one()],             # indirect doctest
...             coerce=False)
>>> I  # random
Left Ideal (y^2, x^2, z^2 - 1) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
>>> sorted(I.gens(), key=str)
[x^2, y^2, z^2 - 1]
>>> H.ideal([y**Integer(2), x**Integer(2), z**Integer(2) - H.one()], side='twosided')  # random
Twosided Ideal (y^2, x^2, z^2 - 1) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
>>> sorted(H.ideal([y**Integer(2), x**Integer(2), z**Integer(2) - H.one()], side='twosided').gens(),
...        key=str)
[x^2, y^2, z^2 - 1]
>>> H.ideal([y**Integer(2), x**Integer(2), z**Integer(2) - H.one()], side='right')
Traceback (most recent call last):
...
ValueError: Only left and two-sided ideals are allowed.
elimination_ideal(variables)[source]

Return the elimination ideal of this ideal with respect to the variables given in variables.

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H = A.g_algebra({y*x: x*y-z, z*x: x*z+2*x, z*y: y*z-2*y})
sage: H.inject_variables()
Defining x, y, z
sage: I = H.ideal([y^2, x^2, z^2 - H.one()], coerce=False)
sage: I.elimination_ideal([x, z])
Left Ideal (y^2) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {...}
sage: J = I.twostd(); J
Twosided Ideal (z^2 - 1, y*z - y, x*z + x, y^2, 2*x*y - z - 1, x^2) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {...}
sage: J.elimination_ideal([x, z])
Twosided Ideal (y^2) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {...}

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> H = A.g_algebra({y*x: x*y-z, z*x: x*z+Integer(2)*x, z*y: y*z-Integer(2)*y})
>>> H.inject_variables()
Defining x, y, z
>>> I = H.ideal([y**Integer(2), x**Integer(2), z**Integer(2) - H.one()], coerce=False)
>>> I.elimination_ideal([x, z])
Left Ideal (y^2) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {...}
>>> J = I.twostd(); J
Twosided Ideal (z^2 - 1, y*z - y, x*z + x, y^2, 2*x*y - z - 1, x^2) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {...}
>>> J.elimination_ideal([x, z])
Twosided Ideal (y^2) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {...}

ALGORITHM: Uses Singular’s eliminate command

reduce(p)[source]

Reduce an element modulo a Groebner basis for this ideal.

It returns 0 if and only if the element is in this ideal. In any case, this reduction is unique up to monomial orders.

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H.<x,y,z> = A.g_algebra({y*x: x*y-z, z*x: x*z+2*x, z*y: y*z-2*y})
sage: I = H.ideal([y^2, x^2, z^2 - H.one()],
....:             coerce=False, side='twosided')
sage: Q = H.quotient(I); Q  #random
Quotient of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
 by the ideal (y^2, x^2, z^2 - 1)
sage: Q.2^2 == Q.one()   # indirect doctest
True

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> H = A.g_algebra({y*x: x*y-z, z*x: x*z+Integer(2)*x, z*y: y*z-Integer(2)*y}, names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> I = H.ideal([y**Integer(2), x**Integer(2), z**Integer(2) - H.one()],
...             coerce=False, side='twosided')
>>> Q = H.quotient(I); Q  #random
Quotient of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
 by the ideal (y^2, x^2, z^2 - 1)
>>> Q.gen(2)**Integer(2) == Q.one()   # indirect doctest
True

Here, we see that the relation that we just found in the quotient is actually a consequence of the given relations:

sage: H.2^2 - H.one() in I.std().gens()                                     # needs sage.combinat sage.modules
True

>>> from sage.all import *
>>> H.gen(2)**Integer(2) - H.one() in I.std().gens()                                     # needs sage.combinat sage.modules
True

Here is the corresponding direct test:

sage: I.reduce(z^2)                                                         # needs sage.combinat sage.modules
1

>>> from sage.all import *
>>> I.reduce(z**Integer(2))                                                         # needs sage.combinat sage.modules
1
res(length)[source]

Compute the resolution up to a given length of the ideal.

NOTE:

Only left syzygies can be computed. So, even if the ideal is two-sided, then the resolution is only one-sided. In that case, a warning is printed.

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H = A.g_algebra({y*x: x*y-z, z*x: x*z+2*x, z*y: y*z-2*y})
sage: H.inject_variables()
Defining x, y, z
sage: I = H.ideal([y^2, x^2, z^2-H.one()], coerce=False)
sage: I.res(3)
<Resolution>

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> H = A.g_algebra({y*x: x*y-z, z*x: x*z+Integer(2)*x, z*y: y*z-Integer(2)*y})
>>> H.inject_variables()
Defining x, y, z
>>> I = H.ideal([y**Integer(2), x**Integer(2), z**Integer(2)-H.one()], coerce=False)
>>> I.res(Integer(3))
<Resolution>
std()[source]

Compute a GB of the ideal. It is two-sided if and only if the ideal is two-sided.

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H = A.g_algebra({y*x: x*y-z, z*x: x*z+2*x, z*y: y*z-2*y})
sage: H.inject_variables()
Defining x, y, z
sage: I = H.ideal([y^2, x^2, z^2 - H.one()], coerce=False)
sage: I.std()  #random
Left Ideal (z^2 - 1, y*z - y, x*z + x, y^2, 2*x*y - z - 1, x^2) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
sage: sorted(I.std().gens(), key=str)
[2*x*y - z - 1, x*z + x, x^2, y*z - y, y^2, z^2 - 1]

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> H = A.g_algebra({y*x: x*y-z, z*x: x*z+Integer(2)*x, z*y: y*z-Integer(2)*y})
>>> H.inject_variables()
Defining x, y, z
>>> I = H.ideal([y**Integer(2), x**Integer(2), z**Integer(2) - H.one()], coerce=False)
>>> I.std()  #random
Left Ideal (z^2 - 1, y*z - y, x*z + x, y^2, 2*x*y - z - 1, x^2) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
>>> sorted(I.std().gens(), key=str)
[2*x*y - z - 1, x*z + x, x^2, y*z - y, y^2, z^2 - 1]

If the ideal is a left ideal, then std() returns a left Groebner basis. But if it is a two-sided ideal, then the output of std() and twostd() coincide:

sage: # needs sage.combinat sage.modules
sage: JL = H.ideal([x^3, y^3, z^3 - 4*z])
sage: JL  #random
Left Ideal (x^3, y^3, z^3 - 4*z) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
sage: sorted(JL.gens(), key=str)
[x^3, y^3, z^3 - 4*z]
sage: JL.std()  # random
Left Ideal (z^3 - 4*z, y*z^2 - 2*y*z,
            x*z^2 + 2*x*z, 2*x*y*z - z^2 - 2*z, y^3, x^3) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
sage: sorted(JL.std().gens(), key=str)
[2*x*y*z - z^2 - 2*z, x*z^2 + 2*x*z, x^3, y*z^2 - 2*y*z, y^3, z^3 - 4*z]
sage: JT = H.ideal([x^3, y^3, z^3 - 4*z], side='twosided')
sage: JT  #random
Twosided Ideal (x^3, y^3, z^3 - 4*z) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
sage: sorted(JT.gens(), key=str)
[x^3, y^3, z^3 - 4*z]
sage: JT.std()  #random
Twosided Ideal (z^3 - 4*z, y*z^2 - 2*y*z, x*z^2 + 2*x*z,
                y^2*z - 2*y^2, 2*x*y*z - z^2 - 2*z, x^2*z + 2*x^2,
                y^3, x*y^2 - y*z, x^2*y - x*z - 2*x, x^3) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
sage: sorted(JT.std().gens(), key=str)
[2*x*y*z - z^2 - 2*z, x*y^2 - y*z, x*z^2 + 2*x*z, x^2*y - x*z - 2*x,
 x^2*z + 2*x^2, x^3, y*z^2 - 2*y*z, y^2*z - 2*y^2, y^3, z^3 - 4*z]
sage: JT.std() == JL.twostd()
True

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> JL = H.ideal([x**Integer(3), y**Integer(3), z**Integer(3) - Integer(4)*z])
>>> JL  #random
Left Ideal (x^3, y^3, z^3 - 4*z) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
>>> sorted(JL.gens(), key=str)
[x^3, y^3, z^3 - 4*z]
>>> JL.std()  # random
Left Ideal (z^3 - 4*z, y*z^2 - 2*y*z,
            x*z^2 + 2*x*z, 2*x*y*z - z^2 - 2*z, y^3, x^3) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
>>> sorted(JL.std().gens(), key=str)
[2*x*y*z - z^2 - 2*z, x*z^2 + 2*x*z, x^3, y*z^2 - 2*y*z, y^3, z^3 - 4*z]
>>> JT = H.ideal([x**Integer(3), y**Integer(3), z**Integer(3) - Integer(4)*z], side='twosided')
>>> JT  #random
Twosided Ideal (x^3, y^3, z^3 - 4*z) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
>>> sorted(JT.gens(), key=str)
[x^3, y^3, z^3 - 4*z]
>>> JT.std()  #random
Twosided Ideal (z^3 - 4*z, y*z^2 - 2*y*z, x*z^2 + 2*x*z,
                y^2*z - 2*y^2, 2*x*y*z - z^2 - 2*z, x^2*z + 2*x^2,
                y^3, x*y^2 - y*z, x^2*y - x*z - 2*x, x^3) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field,
 nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
>>> sorted(JT.std().gens(), key=str)
[2*x*y*z - z^2 - 2*z, x*y^2 - y*z, x*z^2 + 2*x*z, x^2*y - x*z - 2*x,
 x^2*z + 2*x^2, x^3, y*z^2 - 2*y*z, y^2*z - 2*y^2, y^3, z^3 - 4*z]
>>> JT.std() == JL.twostd()
True

ALGORITHM: Uses Singular’s std command

syzygy_module()[source]

Compute the first syzygy (i.e., the module of relations of the given generators) of the ideal.

NOTE:

Only left syzygies can be computed. So, even if the ideal is two-sided, then the syzygies are only one-sided. In that case, a warning is printed.

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H = A.g_algebra({y*x: x*y-z, z*x: x*z+2*x, z*y: y*z-2*y})
sage: H.inject_variables()
Defining x, y, z
sage: I = H.ideal([y^2, x^2, z^2-H.one()], coerce=False)
sage: G = vector(I.gens()); G
d...: UserWarning: You are constructing a free module
over a noncommutative ring. Sage does not have a concept
of left/right and both sided modules, so be careful.
It's also not guaranteed that all multiplications are
done from the right side.
d...: UserWarning: You are constructing a free module
over a noncommutative ring. Sage does not have a concept
of left/right and both sided modules, so be careful.
It's also not guaranteed that all multiplications are
done from the right side.
(y^2, x^2, z^2 - 1)
sage: M = I.syzygy_module(); M
[                                                                         -z^2 - 8*z - 15                                                                                        0                                                                                      y^2]
[                                                                                       0                                                                          -z^2 + 8*z - 15                                                                                      x^2]
[                                                              x^2*z^2 + 8*x^2*z + 15*x^2                                                              -y^2*z^2 + 8*y^2*z - 15*y^2                                                                   -4*x*y*z + 2*z^2 + 2*z]
[                 x^2*y*z^2 + 9*x^2*y*z - 6*x*z^3 + 20*x^2*y - 72*x*z^2 - 282*x*z - 360*x                                                              -y^3*z^2 + 7*y^3*z - 12*y^3                                                                                  6*y*z^2]
[                                                              x^3*z^2 + 7*x^3*z + 12*x^3                 -x*y^2*z^2 + 9*x*y^2*z - 4*y*z^3 - 20*x*y^2 + 52*y*z^2 - 224*y*z + 320*y                                                                                 -6*x*z^2]
[  x^2*y^2*z + 4*x^2*y^2 - 8*x*y*z^2 - 48*x*y*z + 12*z^3 - 64*x*y + 108*z^2 + 312*z + 288                                                                           -y^4*z + 4*y^4                                                                                        0]
[                                                  2*x^3*y*z + 8*x^3*y + 9*x^2*z + 27*x^2                                   -2*x*y^3*z + 8*x*y^3 - 12*y^2*z^2 + 99*y^2*z - 195*y^2                                                                -36*x*y*z + 24*z^2 + 18*z]
[                                                                           x^4*z + 4*x^4    -x^2*y^2*z + 4*x^2*y^2 - 4*x*y*z^2 + 32*x*y*z - 6*z^3 - 64*x*y + 66*z^2 - 240*z + 288                                                                                        0]
[x^3*y^2*z + 4*x^3*y^2 + 18*x^2*y*z - 36*x*z^3 + 66*x^2*y - 432*x*z^2 - 1656*x*z - 2052*x                                      -x*y^4*z + 4*x*y^4 - 8*y^3*z^2 + 62*y^3*z - 114*y^3                                                                        48*y*z^2 - 36*y*z]

sage: M*G                                                                   # needs sage.combinat sage.modules
(0, 0, 0, 0, 0, 0, 0, 0, 0)

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> H = A.g_algebra({y*x: x*y-z, z*x: x*z+Integer(2)*x, z*y: y*z-Integer(2)*y})
>>> H.inject_variables()
Defining x, y, z
>>> I = H.ideal([y**Integer(2), x**Integer(2), z**Integer(2)-H.one()], coerce=False)
>>> G = vector(I.gens()); G
d...: UserWarning: You are constructing a free module
over a noncommutative ring. Sage does not have a concept
of left/right and both sided modules, so be careful.
It's also not guaranteed that all multiplications are
done from the right side.
d...: UserWarning: You are constructing a free module
over a noncommutative ring. Sage does not have a concept
of left/right and both sided modules, so be careful.
It's also not guaranteed that all multiplications are
done from the right side.
(y^2, x^2, z^2 - 1)
>>> M = I.syzygy_module(); M
[                                                                         -z^2 - 8*z - 15                                                                                        0                                                                                      y^2]
[                                                                                       0                                                                          -z^2 + 8*z - 15                                                                                      x^2]
[                                                              x^2*z^2 + 8*x^2*z + 15*x^2                                                              -y^2*z^2 + 8*y^2*z - 15*y^2                                                                   -4*x*y*z + 2*z^2 + 2*z]
[                 x^2*y*z^2 + 9*x^2*y*z - 6*x*z^3 + 20*x^2*y - 72*x*z^2 - 282*x*z - 360*x                                                              -y^3*z^2 + 7*y^3*z - 12*y^3                                                                                  6*y*z^2]
[                                                              x^3*z^2 + 7*x^3*z + 12*x^3                 -x*y^2*z^2 + 9*x*y^2*z - 4*y*z^3 - 20*x*y^2 + 52*y*z^2 - 224*y*z + 320*y                                                                                 -6*x*z^2]
[  x^2*y^2*z + 4*x^2*y^2 - 8*x*y*z^2 - 48*x*y*z + 12*z^3 - 64*x*y + 108*z^2 + 312*z + 288                                                                           -y^4*z + 4*y^4                                                                                        0]
[                                                  2*x^3*y*z + 8*x^3*y + 9*x^2*z + 27*x^2                                   -2*x*y^3*z + 8*x*y^3 - 12*y^2*z^2 + 99*y^2*z - 195*y^2                                                                -36*x*y*z + 24*z^2 + 18*z]
[                                                                           x^4*z + 4*x^4    -x^2*y^2*z + 4*x^2*y^2 - 4*x*y*z^2 + 32*x*y*z - 6*z^3 - 64*x*y + 66*z^2 - 240*z + 288                                                                                        0]
[x^3*y^2*z + 4*x^3*y^2 + 18*x^2*y*z - 36*x*z^3 + 66*x^2*y - 432*x*z^2 - 1656*x*z - 2052*x                                      -x*y^4*z + 4*x*y^4 - 8*y^3*z^2 + 62*y^3*z - 114*y^3                                                                        48*y*z^2 - 36*y*z]

>>> M*G                                                                   # needs sage.combinat sage.modules
(0, 0, 0, 0, 0, 0, 0, 0, 0)

ALGORITHM: Uses Singular’s syz command

twostd()[source]

Compute a two-sided GB of the ideal (even if it is a left ideal).

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H = A.g_algebra({y*x: x*y-z, z*x: x*z+2*x, z*y: y*z-2*y})
sage: H.inject_variables()
Defining x, y, z
sage: I = H.ideal([y^2, x^2, z^2 - H.one()], coerce=False)
sage: I.twostd()  #random
Twosided Ideal (z^2 - 1, y*z - y, x*z + x, y^2, 2*x*y - z - 1, x^2) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field...
sage: sorted(I.twostd().gens(), key=str)
[2*x*y - z - 1, x*z + x, x^2, y*z - y, y^2, z^2 - 1]

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> H = A.g_algebra({y*x: x*y-z, z*x: x*z+Integer(2)*x, z*y: y*z-Integer(2)*y})
>>> H.inject_variables()
Defining x, y, z
>>> I = H.ideal([y**Integer(2), x**Integer(2), z**Integer(2) - H.one()], coerce=False)
>>> I.twostd()  #random
Twosided Ideal (z^2 - 1, y*z - y, x*z + x, y^2, 2*x*y - z - 1, x^2) of
 Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field...
>>> sorted(I.twostd().gens(), key=str)
[2*x*y - z - 1, x*z + x, x^2, y*z - y, y^2, z^2 - 1]

ALGORITHM: Uses Singular’s twostd command

class sage.rings.polynomial.multi_polynomial_ideal.RequireField(f)[source]

Bases: MethodDecorator

Decorator which throws an exception if a computation over a coefficient ring which is not a field is attempted.

Note

This decorator is used automatically internally so the user does not need to use it manually.

sage.rings.polynomial.multi_polynomial_ideal.is_MPolynomialIdeal(x)[source]

Return True if the provided argument x is an ideal in a multivariate polynomial ring.

INPUT:

  • x – an arbitrary object

EXAMPLES:

sage: from sage.rings.polynomial.multi_polynomial_ideal import is_MPolynomialIdeal
sage: P.<x,y,z> = PolynomialRing(QQ)
sage: I = [x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y + 2*y*z - y]

>>> from sage.all import *
>>> from sage.rings.polynomial.multi_polynomial_ideal import is_MPolynomialIdeal
>>> P = PolynomialRing(QQ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> I = [x + Integer(2)*y + Integer(2)*z - Integer(1), x**Integer(2) + Integer(2)*y**Integer(2) + Integer(2)*z**Integer(2) - x, Integer(2)*x*y + Integer(2)*y*z - y]

Sage distinguishes between a list of generators for an ideal and the ideal itself. This distinction is inconsistent with Singular but matches Magma’s behavior.

sage: is_MPolynomialIdeal(I)
doctest:warning...
DeprecationWarning: The function is_MPolynomialIdeal is deprecated;
use 'isinstance(..., MPolynomialIdeal)' instead.
See https://github.com/sagemath/sage/issues/38266 for details.
False

>>> from sage.all import *
>>> is_MPolynomialIdeal(I)
doctest:warning...
DeprecationWarning: The function is_MPolynomialIdeal is deprecated;
use 'isinstance(..., MPolynomialIdeal)' instead.
See https://github.com/sagemath/sage/issues/38266 for details.
False


sage: I = Ideal(I)
sage: is_MPolynomialIdeal(I)
True

>>> from sage.all import *
>>> I = Ideal(I)
>>> is_MPolynomialIdeal(I)
True
sage.rings.polynomial.multi_polynomial_ideal.require_field[source]

alias of RequireField