Generic Multivariate Polynomials

AUTHORS:

  • David Joyner: first version

  • William Stein: use dict’s instead of lists

  • Martin Albrecht malb@informatik.uni-bremen.de: some functions added

  • William Stein (2006-02-11): added better __div__ behavior.

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

  • William Stein (2006-04-19): added e.g., f[1,3] to get coeff of \(xy^3\); added examples of the new R.x,y = PolynomialRing(QQ,2) notation.

  • Martin Albrecht: improved singular coercions (restructured class hierarchy) and added ETuples

  • Robert Bradshaw (2007-08-14): added support for coercion of polynomials in a subset of variables (including multi-level univariate rings)

  • Joel B. Mohler (2008-03): Refactored interactions with ETuples.

EXAMPLES:

We verify Lagrange’s four squares identity:

sage: R.<a0,a1,a2,a3,b0,b1,b2,b3> = QQbar[]                                         # needs sage.rings.number_field
sage: ((a0^2 + a1^2 + a2^2 + a3^2) * (b0^2 + b1^2 + b2^2 + b3^2) ==                 # needs sage.rings.number_field
....:  (a0*b0 - a1*b1 - a2*b2 - a3*b3)^2 + (a0*b1 + a1*b0 + a2*b3 - a3*b2)^2
....:  + (a0*b2 - a1*b3 + a2*b0 + a3*b1)^2 + (a0*b3 + a1*b2 - a2*b1 + a3*b0)^2)
True

>>> from sage.all import *
>>> R = QQbar['a0, a1, a2, a3, b0, b1, b2, b3']; (a0, a1, a2, a3, b0, b1, b2, b3,) = R._first_ngens(8)# needs sage.rings.number_field
>>> ((a0**Integer(2) + a1**Integer(2) + a2**Integer(2) + a3**Integer(2)) * (b0**Integer(2) + b1**Integer(2) + b2**Integer(2) + b3**Integer(2)) ==                 # needs sage.rings.number_field
...  (a0*b0 - a1*b1 - a2*b2 - a3*b3)**Integer(2) + (a0*b1 + a1*b0 + a2*b3 - a3*b2)**Integer(2)
...  + (a0*b2 - a1*b3 + a2*b0 + a3*b1)**Integer(2) + (a0*b3 + a1*b2 - a2*b1 + a3*b0)**Integer(2))
True
class sage.rings.polynomial.multi_polynomial_element.MPolynomial_element(parent, x)[source]

Bases: MPolynomial

Generic multivariate polynomial.

This implementation is based on the PolyDict.

Todo

As mentioned in their docstring, PolyDict objects never clear zeros. In all arithmetic operations on MPolynomial_element there is an additional call to the method remove_zeros to clear them. This is not ideal because of the presence of inexact zeros, see Issue #35174.

element()[source]
hamming_weight()[source]

Return the number of nonzero coefficients of this polynomial.

This is also called weight, hamming_weight() or sparsity.

EXAMPLES:

sage: # needs sage.rings.real_mpfr
sage: R.<x, y> = CC[]
sage: f = x^3 - y
sage: f.number_of_terms()
2
sage: R(0).number_of_terms()
0
sage: f = (x+y)^100
sage: f.number_of_terms()
101

>>> from sage.all import *
>>> # needs sage.rings.real_mpfr
>>> R = CC['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x**Integer(3) - y
>>> f.number_of_terms()
2
>>> R(Integer(0)).number_of_terms()
0
>>> f = (x+y)**Integer(100)
>>> f.number_of_terms()
101

The method hamming_weight() is an alias:

sage: f.hamming_weight()                                                    # needs sage.rings.real_mpfr
101

>>> from sage.all import *
>>> f.hamming_weight()                                                    # needs sage.rings.real_mpfr
101
number_of_terms()[source]

Return the number of nonzero coefficients of this polynomial.

This is also called weight, hamming_weight() or sparsity.

EXAMPLES:

sage: # needs sage.rings.real_mpfr
sage: R.<x, y> = CC[]
sage: f = x^3 - y
sage: f.number_of_terms()
2
sage: R(0).number_of_terms()
0
sage: f = (x+y)^100
sage: f.number_of_terms()
101

>>> from sage.all import *
>>> # needs sage.rings.real_mpfr
>>> R = CC['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x**Integer(3) - y
>>> f.number_of_terms()
2
>>> R(Integer(0)).number_of_terms()
0
>>> f = (x+y)**Integer(100)
>>> f.number_of_terms()
101

The method hamming_weight() is an alias:

sage: f.hamming_weight()                                                    # needs sage.rings.real_mpfr
101

>>> from sage.all import *
>>> f.hamming_weight()                                                    # needs sage.rings.real_mpfr
101
class sage.rings.polynomial.multi_polynomial_element.MPolynomial_polydict(parent, x)[source]

Bases: Polynomial_singular_repr, MPolynomial_element

Multivariate polynomials implemented in pure python using polydicts.

coefficient(degrees)[source]

Return the coefficient of the variables with the degrees specified in the python dictionary degrees. Mathematically, this is the coefficient in the base ring adjoined by the variables of this ring not listed in degrees. However, the result has the same parent as this polynomial.

This function contrasts with the function monomial_coefficient which returns the coefficient in the base ring of a monomial.

INPUT:

  • degrees – can be any of:

    • a dictionary of degree restrictions

    • a list of degree restrictions (with None in the unrestricted variables)

    • a monomial (very fast, but not as flexible)

OUTPUT: element of the parent of self

See also

For coefficients of specific monomials, look at monomial_coefficient().

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x, y> = QQbar[]
sage: f = 2 * x * y
sage: c = f.coefficient({x: 1, y: 1}); c
2
sage: c.parent()
Multivariate Polynomial Ring in x, y over Algebraic Field
sage: c in PolynomialRing(QQbar, 2, names=['x', 'y'])
True
sage: f = y^2 - x^9 - 7*x + 5*x*y
sage: f.coefficient({y: 1})
5*x
sage: f.coefficient({y: 0})
-x^9 + (-7)*x
sage: f.coefficient({x: 0, y: 0})
0
sage: f = (1+y+y^2) * (1+x+x^2)
sage: f.coefficient({x: 0})
y^2 + y + 1
sage: f.coefficient([0, None])
y^2 + y + 1
sage: f.coefficient(x)
y^2 + y + 1
sage: # Be aware that this may not be what you think!
sage: # The physical appearance of the variable x is deceiving -- particularly if the exponent would be a variable.
sage: f.coefficient(x^0) # outputs the full polynomial
x^2*y^2 + x^2*y + x*y^2 + x^2 + x*y + y^2 + x + y + 1

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)
>>> f = Integer(2) * x * y
>>> c = f.coefficient({x: Integer(1), y: Integer(1)}); c
2
>>> c.parent()
Multivariate Polynomial Ring in x, y over Algebraic Field
>>> c in PolynomialRing(QQbar, Integer(2), names=['x', 'y'])
True
>>> f = y**Integer(2) - x**Integer(9) - Integer(7)*x + Integer(5)*x*y
>>> f.coefficient({y: Integer(1)})
5*x
>>> f.coefficient({y: Integer(0)})
-x^9 + (-7)*x
>>> f.coefficient({x: Integer(0), y: Integer(0)})
0
>>> f = (Integer(1)+y+y**Integer(2)) * (Integer(1)+x+x**Integer(2))
>>> f.coefficient({x: Integer(0)})
y^2 + y + 1
>>> f.coefficient([Integer(0), None])
y^2 + y + 1
>>> f.coefficient(x)
y^2 + y + 1
>>> # Be aware that this may not be what you think!
>>> # The physical appearance of the variable x is deceiving -- particularly if the exponent would be a variable.
>>> f.coefficient(x**Integer(0)) # outputs the full polynomial
x^2*y^2 + x^2*y + x*y^2 + x^2 + x*y + y^2 + x + y + 1


sage: # needs sage.rings.real_mpfr
sage: R.<x,y> = RR[]
sage: f = x*y + 5
sage: c = f.coefficient({x: 0, y: 0}); c
5.00000000000000
sage: parent(c)
Multivariate Polynomial Ring in x, y over Real Field with 53 bits of precision

>>> from sage.all import *
>>> # needs sage.rings.real_mpfr
>>> R = RR['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x*y + Integer(5)
>>> c = f.coefficient({x: Integer(0), y: Integer(0)}); c
5.00000000000000
>>> parent(c)
Multivariate Polynomial Ring in x, y over Real Field with 53 bits of precision

AUTHORS:

  • Joel B. Mohler (2007-10-31)

constant_coefficient()[source]

Return the constant coefficient of this multivariate polynomial.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y> = QQbar[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.constant_coefficient()
5
sage: f = 3*x^2
sage: f.constant_coefficient()
0

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)
>>> f = Integer(3)*x**Integer(2) - Integer(2)*y + Integer(7)*x**Integer(2)*y**Integer(2) + Integer(5)
>>> f.constant_coefficient()
5
>>> f = Integer(3)*x**Integer(2)
>>> f.constant_coefficient()
0
degree(x=None, std_grading=False)[source]

Return the degree of self in x, where x must be one of the generators for the parent of self.

INPUT:

  • x – multivariate polynomial (a generator of the parent of self). If x is not specified (or is None), return the total degree, which is the maximum degree of any monomial. Note that a weighted term ordering alters the grading of the generators of the ring; see the tests below. To avoid this behavior, set the optional argument std_grading=True.

OUTPUT: integer

EXAMPLES:

sage: R.<x,y> = RR[]
sage: f = y^2 - x^9 - x
sage: f.degree(x)
9
sage: f.degree(y)
2
sage: (y^10*x - 7*x^2*y^5 + 5*x^3).degree(x)
3
sage: (y^10*x - 7*x^2*y^5 + 5*x^3).degree(y)
10

>>> from sage.all import *
>>> R = RR['x, y']; (x, y,) = R._first_ngens(2)
>>> f = y**Integer(2) - x**Integer(9) - x
>>> f.degree(x)
9
>>> f.degree(y)
2
>>> (y**Integer(10)*x - Integer(7)*x**Integer(2)*y**Integer(5) + Integer(5)*x**Integer(3)).degree(x)
3
>>> (y**Integer(10)*x - Integer(7)*x**Integer(2)*y**Integer(5) + Integer(5)*x**Integer(3)).degree(y)
10

Note that total degree takes into account if we are working in a polynomial ring with a weighted term order.

sage: R = PolynomialRing(QQ, 'x,y', order=TermOrder('wdeglex',(2,3)))
sage: x,y = R.gens()
sage: x.degree()
2
sage: y.degree()
3
sage: x.degree(y), x.degree(x), y.degree(x), y.degree(y)
(0, 1, 0, 1)
sage: f = x^2*y + x*y^2
sage: f.degree(x)
2
sage: f.degree(y)
2
sage: f.degree()
8
sage: f.degree(std_grading=True)
3

>>> from sage.all import *
>>> R = PolynomialRing(QQ, 'x,y', order=TermOrder('wdeglex',(Integer(2),Integer(3))))
>>> x,y = R.gens()
>>> x.degree()
2
>>> y.degree()
3
>>> x.degree(y), x.degree(x), y.degree(x), y.degree(y)
(0, 1, 0, 1)
>>> f = x**Integer(2)*y + x*y**Integer(2)
>>> f.degree(x)
2
>>> f.degree(y)
2
>>> f.degree()
8
>>> f.degree(std_grading=True)
3

Note that if x is not a generator of the parent of self, for example if it is a generator of a polynomial algebra which maps naturally to this one, then it is converted to an element of this algebra. (This fixes the problem reported in Issue #17366.)

sage: x, y = ZZ['x','y'].gens()
sage: GF(3037000453)['x','y'].gen(0).degree(x)                              # needs sage.rings.finite_rings
1

sage: x0, y0 = QQ['x','y'].gens()
sage: GF(3037000453)['x','y'].gen(0).degree(x0)                             # needs sage.rings.finite_rings
Traceback (most recent call last):
...
TypeError: x must canonically coerce to parent

sage: GF(3037000453)['x','y'].gen(0).degree(x^2)                            # needs sage.rings.finite_rings
Traceback (most recent call last):
...
TypeError: x must be one of the generators of the parent

>>> from sage.all import *
>>> x, y = ZZ['x','y'].gens()
>>> GF(Integer(3037000453))['x','y'].gen(Integer(0)).degree(x)                              # needs sage.rings.finite_rings
1

>>> x0, y0 = QQ['x','y'].gens()
>>> GF(Integer(3037000453))['x','y'].gen(Integer(0)).degree(x0)                             # needs sage.rings.finite_rings
Traceback (most recent call last):
...
TypeError: x must canonically coerce to parent

>>> GF(Integer(3037000453))['x','y'].gen(Integer(0)).degree(x**Integer(2))                            # needs sage.rings.finite_rings
Traceback (most recent call last):
...
TypeError: x must be one of the generators of the parent
degrees()[source]

Return a tuple (precisely - an ETuple) with the degree of each variable in this polynomial. The list of degrees is, of course, ordered by the order of the generators.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y,z> = PolynomialRing(QQbar)
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.degrees()
(2, 2, 0)
sage: f = x^2 + z^2
sage: f.degrees()
(2, 0, 2)
sage: f.total_degree()  # this simply illustrates that total degree is not the sum of the degrees
2
sage: R.<x,y,z,u> = PolynomialRing(QQbar)
sage: f = (1-x) * (1+y+z+x^3)^5
sage: f.degrees()
(16, 5, 5, 0)
sage: R(0).degrees()
(0, 0, 0, 0)

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = PolynomialRing(QQbar, names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> f = Integer(3)*x**Integer(2) - Integer(2)*y + Integer(7)*x**Integer(2)*y**Integer(2) + Integer(5)
>>> f.degrees()
(2, 2, 0)
>>> f = x**Integer(2) + z**Integer(2)
>>> f.degrees()
(2, 0, 2)
>>> f.total_degree()  # this simply illustrates that total degree is not the sum of the degrees
2
>>> R = PolynomialRing(QQbar, names=('x', 'y', 'z', 'u',)); (x, y, z, u,) = R._first_ngens(4)
>>> f = (Integer(1)-x) * (Integer(1)+y+z+x**Integer(3))**Integer(5)
>>> f.degrees()
(16, 5, 5, 0)
>>> R(Integer(0)).degrees()
(0, 0, 0, 0)
dict()[source]

Return underlying dictionary with keys the exponents and values the coefficients of this polynomial.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y,z> = PolynomialRing(QQbar, order='lex')
sage: f = (x^1*y^5*z^2 + x^2*z + x^4*y^1*z^3)
sage: f.monomial_coefficients()
{(1, 5, 2): 1, (2, 0, 1): 1, (4, 1, 3): 1}

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = PolynomialRing(QQbar, order='lex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> f = (x**Integer(1)*y**Integer(5)*z**Integer(2) + x**Integer(2)*z + x**Integer(4)*y**Integer(1)*z**Integer(3))
>>> f.monomial_coefficients()
{(1, 5, 2): 1, (2, 0, 1): 1, (4, 1, 3): 1}

dict is an alias:

sage: f.dict()  # needs sage.rings.number_field
{(1, 5, 2): 1, (2, 0, 1): 1, (4, 1, 3): 1}

>>> from sage.all import *
>>> f.dict()  # needs sage.rings.number_field
{(1, 5, 2): 1, (2, 0, 1): 1, (4, 1, 3): 1}
exponents(as_ETuples=True)[source]

Return the exponents of the monomials appearing in self.

INPUT:

  • as_ETuples – (default: True) return the list of exponents as a list of ETuples

OUTPUT: the list of exponents as a list of ETuples or tuples

EXAMPLES:

sage: R.<a,b,c> = PolynomialRing(QQbar, 3)                                  # needs sage.rings.number_field
sage: f = a^3 + b + 2*b^2                                                   # needs sage.rings.number_field
sage: f.exponents()                                                         # needs sage.rings.number_field
[(3, 0, 0), (0, 2, 0), (0, 1, 0)]

>>> from sage.all import *
>>> R = PolynomialRing(QQbar, Integer(3), names=('a', 'b', 'c',)); (a, b, c,) = R._first_ngens(3)# needs sage.rings.number_field
>>> f = a**Integer(3) + b + Integer(2)*b**Integer(2)                                                   # needs sage.rings.number_field
>>> f.exponents()                                                         # needs sage.rings.number_field
[(3, 0, 0), (0, 2, 0), (0, 1, 0)]

By default the list of exponents is a list of ETuples:

sage: type(f.exponents()[0])                                                # needs sage.rings.number_field
<class 'sage.rings.polynomial.polydict.ETuple'>
sage: type(f.exponents(as_ETuples=False)[0])                                # needs sage.rings.number_field
<... 'tuple'>

>>> from sage.all import *
>>> type(f.exponents()[Integer(0)])                                                # needs sage.rings.number_field
<class 'sage.rings.polynomial.polydict.ETuple'>
>>> type(f.exponents(as_ETuples=False)[Integer(0)])                                # needs sage.rings.number_field
<... 'tuple'>
factor(proof=None)[source]

Compute the irreducible factorization of this polynomial.

INPUT:

global_height(prec=None)[source]

Return the (projective) global height of the polynomial.

This returns the absolute logarithmic height of the coefficients thought of as a projective point.

INPUT:

  • prec – desired floating point precision (default: default RealField precision)

OUTPUT: a real number

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQbar, 2)                                    # needs sage.rings.number_field
sage: f = QQbar(i)*x^2 + 3*x*y                                              # needs sage.rings.number_field
sage: f.global_height()                                                     # needs sage.rings.number_field
1.09861228866811

>>> from sage.all import *
>>> R = PolynomialRing(QQbar, Integer(2), names=('x', 'y',)); (x, y,) = R._first_ngens(2)# needs sage.rings.number_field
>>> f = QQbar(i)*x**Integer(2) + Integer(3)*x*y                                              # needs sage.rings.number_field
>>> f.global_height()                                                     # needs sage.rings.number_field
1.09861228866811

Scaling should not change the result:

sage: # needs sage.rings.number_field sage.symbolic
sage: R.<x, y> = PolynomialRing(QQbar, 2)
sage: f = 1/25*x^2 + 25/3*x + 1 + QQbar(sqrt(2))*y^2
sage: f.global_height()
6.43775164973640
sage: g = 100 * f
sage: g.global_height()
6.43775164973640

>>> from sage.all import *
>>> # needs sage.rings.number_field sage.symbolic
>>> R = PolynomialRing(QQbar, Integer(2), names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(1)/Integer(25)*x**Integer(2) + Integer(25)/Integer(3)*x + Integer(1) + QQbar(sqrt(Integer(2)))*y**Integer(2)
>>> f.global_height()
6.43775164973640
>>> g = Integer(100) * f
>>> g.global_height()
6.43775164973640


sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: K.<k> = NumberField(x^2 + 1)
sage: Q.<q,r> = PolynomialRing(K, implementation='generic')
sage: f = 12 * q
sage: f.global_height()
0.000000000000000

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('k',)); (k,) = K._first_ngens(1)
>>> Q = PolynomialRing(K, implementation='generic', names=('q', 'r',)); (q, r,) = Q._first_ngens(2)
>>> f = Integer(12) * q
>>> f.global_height()
0.000000000000000


sage: R.<x,y> = PolynomialRing(QQ, implementation='generic')
sage: f = 1/123*x*y + 12
sage: f.global_height(prec=2)                                               # needs sage.symbolic
8.0

>>> from sage.all import *
>>> R = PolynomialRing(QQ, implementation='generic', names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(1)/Integer(123)*x*y + Integer(12)
>>> f.global_height(prec=Integer(2))                                               # needs sage.symbolic
8.0


sage: R.<x,y> = PolynomialRing(QQ, implementation='generic')
sage: f = 0*x*y
sage: f.global_height()                                                     # needs sage.rings.real_mpfr
0.000000000000000

>>> from sage.all import *
>>> R = PolynomialRing(QQ, implementation='generic', names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(0)*x*y
>>> f.global_height()                                                     # needs sage.rings.real_mpfr
0.000000000000000
integral(var=None)[source]

Integrate self with respect to variable var.

Note

The integral is always chosen so the constant term is 0.

If var is not one of the generators of this ring, integral(var) is called recursively on each coefficient of this polynomial.

EXAMPLES:

On polynomials with rational coefficients:

sage: x, y = PolynomialRing(QQ, 'x, y').gens()
sage: ex = x*y + x - y
sage: it = ex.integral(x); it
1/2*x^2*y + 1/2*x^2 - x*y
sage: it.parent() == x.parent()
True

sage: R = ZZ['x']['y, z']
sage: y, z = R.gens()
sage: R.an_element().integral(y).parent()
Multivariate Polynomial Ring in y, z
 over Univariate Polynomial Ring in x over Rational Field

>>> from sage.all import *
>>> x, y = PolynomialRing(QQ, 'x, y').gens()
>>> ex = x*y + x - y
>>> it = ex.integral(x); it
1/2*x^2*y + 1/2*x^2 - x*y
>>> it.parent() == x.parent()
True

>>> R = ZZ['x']['y, z']
>>> y, z = R.gens()
>>> R.an_element().integral(y).parent()
Multivariate Polynomial Ring in y, z
 over Univariate Polynomial Ring in x over Rational Field

On polynomials with coefficients in power series:

sage: # needs sage.rings.number_field
sage: R.<t> = PowerSeriesRing(QQbar)
sage: S.<x, y> = PolynomialRing(R)
sage: f = (t^2 + O(t^3))*x^2*y^3 + (37*t^4 + O(t^5))*x^3
sage: f.parent()
Multivariate Polynomial Ring in x, y
 over Power Series Ring in t over Algebraic Field
sage: f.integral(x)   # with respect to x
(1/3*t^2 + O(t^3))*x^3*y^3 + (37/4*t^4 + O(t^5))*x^4
sage: f.integral(x).parent()
Multivariate Polynomial Ring in x, y
 over Power Series Ring in t over Algebraic Field
sage: f.integral(y)   # with respect to y
(1/4*t^2 + O(t^3))*x^2*y^4 + (37*t^4 + O(t^5))*x^3*y
sage: f.integral(t)   # with respect to t (recurses into base ring)
(1/3*t^3 + O(t^4))*x^2*y^3 + (37/5*t^5 + O(t^6))*x^3

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = PowerSeriesRing(QQbar, names=('t',)); (t,) = R._first_ngens(1)
>>> S = PolynomialRing(R, names=('x', 'y',)); (x, y,) = S._first_ngens(2)
>>> f = (t**Integer(2) + O(t**Integer(3)))*x**Integer(2)*y**Integer(3) + (Integer(37)*t**Integer(4) + O(t**Integer(5)))*x**Integer(3)
>>> f.parent()
Multivariate Polynomial Ring in x, y
 over Power Series Ring in t over Algebraic Field
>>> f.integral(x)   # with respect to x
(1/3*t^2 + O(t^3))*x^3*y^3 + (37/4*t^4 + O(t^5))*x^4
>>> f.integral(x).parent()
Multivariate Polynomial Ring in x, y
 over Power Series Ring in t over Algebraic Field
>>> f.integral(y)   # with respect to y
(1/4*t^2 + O(t^3))*x^2*y^4 + (37*t^4 + O(t^5))*x^3*y
>>> f.integral(t)   # with respect to t (recurses into base ring)
(1/3*t^3 + O(t^4))*x^2*y^3 + (37/5*t^5 + O(t^6))*x^3
inverse_of_unit()[source]

Return the inverse of a unit in a ring.

is_constant()[source]

Return True if self is a constant and False otherwise.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y> = QQbar[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.is_constant()
False
sage: g = 10*x^0
sage: g.is_constant()
True

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)
>>> f = Integer(3)*x**Integer(2) - Integer(2)*y + Integer(7)*x**Integer(2)*y**Integer(2) + Integer(5)
>>> f.is_constant()
False
>>> g = Integer(10)*x**Integer(0)
>>> g.is_constant()
True
is_gen()[source]

Return True if self is a generator of its parent.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y> = QQbar[]
sage: x.is_gen()
True
sage: (x + y - y).is_gen()
True
sage: (x*y).is_gen()
False

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)
>>> x.is_gen()
True
>>> (x + y - y).is_gen()
True
>>> (x*y).is_gen()
False
is_generator(*args, **kwds)[source]

Deprecated: Use is_gen() instead. See Issue #38942 for details.

is_homogeneous()[source]

Return True if self is a homogeneous polynomial.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y> = QQbar[]
sage: (x + y).is_homogeneous()
True
sage: (x.parent()(0)).is_homogeneous()
True
sage: (x + y^2).is_homogeneous()
False
sage: (x^2 + y^2).is_homogeneous()
True
sage: (x^2 + y^2*x).is_homogeneous()
False
sage: (x^2*y + y^2*x).is_homogeneous()
True

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)
>>> (x + y).is_homogeneous()
True
>>> (x.parent()(Integer(0))).is_homogeneous()
True
>>> (x + y**Integer(2)).is_homogeneous()
False
>>> (x**Integer(2) + y**Integer(2)).is_homogeneous()
True
>>> (x**Integer(2) + y**Integer(2)*x).is_homogeneous()
False
>>> (x**Integer(2)*y + y**Integer(2)*x).is_homogeneous()
True

The weight of the parent ring is respected:

sage: term_order = TermOrder("wdegrevlex", [1, 3])
sage: R.<x, y> = PolynomialRing(Qp(5), order=term_order)
sage: (x + y).is_homogeneous()
False
sage: (x^3 + y).is_homogeneous()
True

>>> from sage.all import *
>>> term_order = TermOrder("wdegrevlex", [Integer(1), Integer(3)])
>>> R = PolynomialRing(Qp(Integer(5)), order=term_order, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> (x + y).is_homogeneous()
False
>>> (x**Integer(3) + y).is_homogeneous()
True
is_monomial()[source]

Return True if self is a monomial, which we define to be a product of generators with coefficient 1.

Use is_term() to allow the coefficient to not be 1.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y> = QQbar[]
sage: x.is_monomial()
True
sage: (x + 2*y).is_monomial()
False
sage: (2*x).is_monomial()
False
sage: (x*y).is_monomial()
True

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)
>>> x.is_monomial()
True
>>> (x + Integer(2)*y).is_monomial()
False
>>> (Integer(2)*x).is_monomial()
False
>>> (x*y).is_monomial()
True

To allow a non-1 leading coefficient, use is_term():

sage: (2*x*y).is_term()                                                     # needs sage.rings.number_field
True
sage: (2*x*y).is_monomial()                                                 # needs sage.rings.number_field
False

>>> from sage.all import *
>>> (Integer(2)*x*y).is_term()                                                     # needs sage.rings.number_field
True
>>> (Integer(2)*x*y).is_monomial()                                                 # needs sage.rings.number_field
False
is_term()[source]

Return True if self is a term, which we define to be a product of generators times some coefficient, which need not be 1.

Use is_monomial() to require that the coefficient be 1.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y> = QQbar[]
sage: x.is_term()
True
sage: (x + 2*y).is_term()
False
sage: (2*x).is_term()
True
sage: (7*x^5*y).is_term()
True

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)
>>> x.is_term()
True
>>> (x + Integer(2)*y).is_term()
False
>>> (Integer(2)*x).is_term()
True
>>> (Integer(7)*x**Integer(5)*y).is_term()
True

To require leading coefficient 1, use is_monomial():

sage: (2*x*y).is_monomial()                                                 # needs sage.rings.number_field
False
sage: (2*x*y).is_term()                                                     # needs sage.rings.number_field
True

>>> from sage.all import *
>>> (Integer(2)*x*y).is_monomial()                                                 # needs sage.rings.number_field
False
>>> (Integer(2)*x*y).is_term()                                                     # needs sage.rings.number_field
True
is_univariate()[source]

Return True if this multivariate polynomial is univariate and False otherwise.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y> = QQbar[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.is_univariate()
False
sage: g = f.subs({x: 10}); g
700*y^2 + (-2)*y + 305
sage: g.is_univariate()
True
sage: f = x^0
sage: f.is_univariate()
True

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)
>>> f = Integer(3)*x**Integer(2) - Integer(2)*y + Integer(7)*x**Integer(2)*y**Integer(2) + Integer(5)
>>> f.is_univariate()
False
>>> g = f.subs({x: Integer(10)}); g
700*y^2 + (-2)*y + 305
>>> g.is_univariate()
True
>>> f = x**Integer(0)
>>> f.is_univariate()
True
iterator_exp_coeff(as_ETuples=True)[source]

Iterate over self as pairs of ((E)Tuple, coefficient).

INPUT:

  • as_ETuples – boolean (default: True); if True iterate over pairs whose first element is an ETuple, otherwise as a tuples

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(QQbar, order='lex')                        # needs sage.rings.number_field
sage: f = (x^1*y^5*z^2 + x^2*z + x^4*y^1*z^3)                               # needs sage.rings.number_field
sage: list(f.iterator_exp_coeff())                                          # needs sage.rings.number_field
[((4, 1, 3), 1), ((2, 0, 1), 1), ((1, 5, 2), 1)]

sage: R.<x,y,z> = PolynomialRing(QQbar, order='deglex')                     # needs sage.rings.number_field
sage: f = (x^1*y^5*z^2 + x^2*z + x^4*y^1*z^3)                               # needs sage.rings.number_field
sage: list(f.iterator_exp_coeff(as_ETuples=False))                          # needs sage.rings.number_field
[((4, 1, 3), 1), ((1, 5, 2), 1), ((2, 0, 1), 1)]

>>> from sage.all import *
>>> R = PolynomialRing(QQbar, order='lex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)# needs sage.rings.number_field
>>> f = (x**Integer(1)*y**Integer(5)*z**Integer(2) + x**Integer(2)*z + x**Integer(4)*y**Integer(1)*z**Integer(3))                               # needs sage.rings.number_field
>>> list(f.iterator_exp_coeff())                                          # needs sage.rings.number_field
[((4, 1, 3), 1), ((2, 0, 1), 1), ((1, 5, 2), 1)]

>>> R = PolynomialRing(QQbar, order='deglex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)# needs sage.rings.number_field
>>> f = (x**Integer(1)*y**Integer(5)*z**Integer(2) + x**Integer(2)*z + x**Integer(4)*y**Integer(1)*z**Integer(3))                               # needs sage.rings.number_field
>>> list(f.iterator_exp_coeff(as_ETuples=False))                          # needs sage.rings.number_field
[((4, 1, 3), 1), ((1, 5, 2), 1), ((2, 0, 1), 1)]
lc()[source]

Return the leading coefficient of self, i.e., self.coefficient(self.lm()).

EXAMPLES:

sage: R.<x,y,z> = QQbar[]                                                   # needs sage.rings.number_field
sage: f = 3*x^2 - y^2 - x*y                                                 # needs sage.rings.number_field
sage: f.lc()                                                                # needs sage.rings.number_field
3

>>> from sage.all import *
>>> R = QQbar['x, y, z']; (x, y, z,) = R._first_ngens(3)# needs sage.rings.number_field
>>> f = Integer(3)*x**Integer(2) - y**Integer(2) - x*y                                                 # needs sage.rings.number_field
>>> f.lc()                                                                # needs sage.rings.number_field
3
lift(I)[source]

Given an ideal \(I = (f_1,...,f_r)\) and some \(g\) (= self) in \(I\), find \(s_1,...,s_r\) such that \(g = s_1 f_1 + ... + s_r f_r\).

ALGORITHM: Use Singular.

EXAMPLES:

sage: # needs sage.rings.real_mpfr
sage: A.<x,y> = PolynomialRing(CC, 2, order='degrevlex')
sage: I = A.ideal([x^10 + x^9*y^2, y^8 - x^2*y^7])
sage: f = x*y^13 + y^12
sage: M = f.lift(I); M                                                      # needs sage.libs.singular
[y^7, x^7*y^2 + x^8 + x^5*y^3 + x^6*y + x^3*y^4 + x^4*y^2 + x*y^5 + x^2*y^3 + y^4]
sage: sum(map(mul, zip(M, I.gens()))) == f                                  # needs sage.libs.singular
True

>>> from sage.all import *
>>> # needs sage.rings.real_mpfr
>>> A = PolynomialRing(CC, Integer(2), order='degrevlex', names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> I = A.ideal([x**Integer(10) + x**Integer(9)*y**Integer(2), y**Integer(8) - x**Integer(2)*y**Integer(7)])
>>> f = x*y**Integer(13) + y**Integer(12)
>>> M = f.lift(I); M                                                      # needs sage.libs.singular
[y^7, x^7*y^2 + x^8 + x^5*y^3 + x^6*y + x^3*y^4 + x^4*y^2 + x*y^5 + x^2*y^3 + y^4]
>>> sum(map(mul, zip(M, I.gens()))) == f                                  # needs sage.libs.singular
True
lm()[source]

Return the lead monomial of self with respect to the term order of self.parent().

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(GF(7), 3, order='lex')
sage: (x^1*y^2 + y^3*z^4).lm()
x*y^2
sage: (x^3*y^2*z^4 + x^3*y^2*z^1).lm()
x^3*y^2*z^4

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(7)), Integer(3), order='lex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> (x**Integer(1)*y**Integer(2) + y**Integer(3)*z**Integer(4)).lm()
x*y^2
>>> (x**Integer(3)*y**Integer(2)*z**Integer(4) + x**Integer(3)*y**Integer(2)*z**Integer(1)).lm()
x^3*y^2*z^4


sage: # needs sage.rings.real_mpfr
sage: R.<x,y,z> = PolynomialRing(CC, 3, order='deglex')
sage: (x^1*y^2*z^3 + x^3*y^2*z^0).lm()
x*y^2*z^3
sage: (x^1*y^2*z^4 + x^1*y^1*z^5).lm()
x*y^2*z^4

>>> from sage.all import *
>>> # needs sage.rings.real_mpfr
>>> R = PolynomialRing(CC, Integer(3), order='deglex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> (x**Integer(1)*y**Integer(2)*z**Integer(3) + x**Integer(3)*y**Integer(2)*z**Integer(0)).lm()
x*y^2*z^3
>>> (x**Integer(1)*y**Integer(2)*z**Integer(4) + x**Integer(1)*y**Integer(1)*z**Integer(5)).lm()
x*y^2*z^4


sage: # needs sage.rings.number_field
sage: R.<x,y,z> = PolynomialRing(QQbar, 3, order='degrevlex')
sage: (x^1*y^5*z^2 + x^4*y^1*z^3).lm()
x*y^5*z^2
sage: (x^4*y^7*z^1 + x^4*y^2*z^3).lm()
x^4*y^7*z

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = PolynomialRing(QQbar, Integer(3), order='degrevlex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> (x**Integer(1)*y**Integer(5)*z**Integer(2) + x**Integer(4)*y**Integer(1)*z**Integer(3)).lm()
x*y^5*z^2
>>> (x**Integer(4)*y**Integer(7)*z**Integer(1) + x**Integer(4)*y**Integer(2)*z**Integer(3)).lm()
x^4*y^7*z
local_height(v, prec=None)[source]

Return the maximum of the local height of the coefficients of this polynomial.

INPUT:

  • v – a prime or prime ideal of the base ring

  • prec – desired floating point precision (default: default RealField precision)

OUTPUT: a real number

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ, implementation='generic')
sage: f = 1/1331*x^2 + 1/4000*y
sage: f.local_height(1331)                                                  # needs sage.rings.real_mpfr
7.19368581839511

>>> from sage.all import *
>>> R = PolynomialRing(QQ, implementation='generic', names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(1)/Integer(1331)*x**Integer(2) + Integer(1)/Integer(4000)*y
>>> f.local_height(Integer(1331))                                                  # needs sage.rings.real_mpfr
7.19368581839511


sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: K.<k> = NumberField(x^2 - 5)
sage: T.<t,w> = PolynomialRing(K, implementation='generic')
sage: I = K.ideal(3)
sage: f = 1/3*t*w + 3
sage: f.local_height(I)                                                     # needs sage.symbolic
1.09861228866811

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(2) - Integer(5), names=('k',)); (k,) = K._first_ngens(1)
>>> T = PolynomialRing(K, implementation='generic', names=('t', 'w',)); (t, w,) = T._first_ngens(2)
>>> I = K.ideal(Integer(3))
>>> f = Integer(1)/Integer(3)*t*w + Integer(3)
>>> f.local_height(I)                                                     # needs sage.symbolic
1.09861228866811


sage: R.<x,y> = PolynomialRing(QQ, implementation='generic')
sage: f = 1/2*x*y + 2
sage: f.local_height(2, prec=2)                                             # needs sage.rings.real_mpfr
0.75

>>> from sage.all import *
>>> R = PolynomialRing(QQ, implementation='generic', names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(1)/Integer(2)*x*y + Integer(2)
>>> f.local_height(Integer(2), prec=Integer(2))                                             # needs sage.rings.real_mpfr
0.75
local_height_arch(i, prec=None)[source]

Return the maximum of the local height at the i-th infinite place of the coefficients of this polynomial.

INPUT:

  • i – integer

  • prec – desired floating point precision (default: default RealField precision)

OUTPUT: a real number

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ, implementation='generic')
sage: f = 210*x*y
sage: f.local_height_arch(0)                                                # needs sage.rings.real_mpfr
5.34710753071747

>>> from sage.all import *
>>> R = PolynomialRing(QQ, implementation='generic', names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(210)*x*y
>>> f.local_height_arch(Integer(0))                                                # needs sage.rings.real_mpfr
5.34710753071747


sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: K.<k> = NumberField(x^2 - 5)
sage: T.<t,w> = PolynomialRing(K, implementation='generic')
sage: f = 1/2*t*w + 3
sage: f.local_height_arch(1, prec=52)
1.09861228866811

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(2) - Integer(5), names=('k',)); (k,) = K._first_ngens(1)
>>> T = PolynomialRing(K, implementation='generic', names=('t', 'w',)); (t, w,) = T._first_ngens(2)
>>> f = Integer(1)/Integer(2)*t*w + Integer(3)
>>> f.local_height_arch(Integer(1), prec=Integer(52))
1.09861228866811


sage: R.<x,y> = PolynomialRing(QQ, implementation='generic')
sage: f = 1/2*x*y + 3
sage: f.local_height_arch(0, prec=2)                                        # needs sage.rings.real_mpfr
1.0

>>> from sage.all import *
>>> R = PolynomialRing(QQ, implementation='generic', names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(1)/Integer(2)*x*y + Integer(3)
>>> f.local_height_arch(Integer(0), prec=Integer(2))                                        # needs sage.rings.real_mpfr
1.0
lt()[source]

Return the leading term of self i.e., self.lc()*self.lm(). The notion of “leading term” depends on the ordering defined in the parent ring.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y,z> = PolynomialRing(QQbar)
sage: f = 3*x^2 - y^2 - x*y
sage: f.lt()
3*x^2
sage: R.<x,y,z> = PolynomialRing(QQbar, order='invlex')
sage: f = 3*x^2 - y^2 - x*y
sage: f.lt()
-y^2

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = PolynomialRing(QQbar, names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> f = Integer(3)*x**Integer(2) - y**Integer(2) - x*y
>>> f.lt()
3*x^2
>>> R = PolynomialRing(QQbar, order='invlex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> f = Integer(3)*x**Integer(2) - y**Integer(2) - x*y
>>> f.lt()
-y^2
monomial_coefficient(mon)[source]

Return the coefficient in the base ring of the monomial mon in self, where mon must have the same parent as self.

This function contrasts with the function coefficient which returns the coefficient of a monomial viewing this polynomial in a polynomial ring over a base ring having fewer variables.

INPUT:

  • mon – a monomial

OUTPUT: coefficient in base ring

See also

For coefficients in a base ring of fewer variables, look at coefficient().

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y> = QQbar[]
sage: f = 2 * x * y
sage: c = f.monomial_coefficient(x*y); c
2
sage: c.parent()
Algebraic Field

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)
>>> f = Integer(2) * x * y
>>> c = f.monomial_coefficient(x*y); c
2
>>> c.parent()
Algebraic Field


sage: # needs sage.rings.number_field
sage: f = y^2 + y^2*x - x^9 - 7*x + 5*x*y
sage: f.monomial_coefficient(y^2)
1
sage: f.monomial_coefficient(x*y)
5
sage: f.monomial_coefficient(x^9)
-1
sage: f.monomial_coefficient(x^10)
0

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> f = y**Integer(2) + y**Integer(2)*x - x**Integer(9) - Integer(7)*x + Integer(5)*x*y
>>> f.monomial_coefficient(y**Integer(2))
1
>>> f.monomial_coefficient(x*y)
5
>>> f.monomial_coefficient(x**Integer(9))
-1
>>> f.monomial_coefficient(x**Integer(10))
0


sage: # needs sage.rings.number_field
sage: a = polygen(ZZ, 'a')
sage: K.<a> = NumberField(a^2 + a + 1)
sage: P.<x,y> = K[]
sage: f = (a*x - 1) * ((a+1)*y - 1); f
-x*y + (-a)*x + (-a - 1)*y + 1
sage: f.monomial_coefficient(x)
-a

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> a = polygen(ZZ, 'a')
>>> K = NumberField(a**Integer(2) + a + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K['x, y']; (x, y,) = P._first_ngens(2)
>>> f = (a*x - Integer(1)) * ((a+Integer(1))*y - Integer(1)); f
-x*y + (-a)*x + (-a - 1)*y + 1
>>> f.monomial_coefficient(x)
-a
monomial_coefficients()[source]

Return underlying dictionary with keys the exponents and values the coefficients of this polynomial.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y,z> = PolynomialRing(QQbar, order='lex')
sage: f = (x^1*y^5*z^2 + x^2*z + x^4*y^1*z^3)
sage: f.monomial_coefficients()
{(1, 5, 2): 1, (2, 0, 1): 1, (4, 1, 3): 1}

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = PolynomialRing(QQbar, order='lex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> f = (x**Integer(1)*y**Integer(5)*z**Integer(2) + x**Integer(2)*z + x**Integer(4)*y**Integer(1)*z**Integer(3))
>>> f.monomial_coefficients()
{(1, 5, 2): 1, (2, 0, 1): 1, (4, 1, 3): 1}

dict is an alias:

sage: f.dict()  # needs sage.rings.number_field
{(1, 5, 2): 1, (2, 0, 1): 1, (4, 1, 3): 1}

>>> from sage.all import *
>>> f.dict()  # needs sage.rings.number_field
{(1, 5, 2): 1, (2, 0, 1): 1, (4, 1, 3): 1}
monomials()[source]

Return the list of monomials in self. The returned list is decreasingly ordered by the term ordering of self.parent().

OUTPUT: list of MPolynomial instances, representing monomials

EXAMPLES:

sage: R.<x,y> = QQbar[]                                                     # needs sage.rings.number_field
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5                                       # needs sage.rings.number_field
sage: f.monomials()                                                         # needs sage.rings.number_field
[x^2*y^2, x^2, y, 1]

>>> from sage.all import *
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)# needs sage.rings.number_field
>>> f = Integer(3)*x**Integer(2) - Integer(2)*y + Integer(7)*x**Integer(2)*y**Integer(2) + Integer(5)                                       # needs sage.rings.number_field
>>> f.monomials()                                                         # needs sage.rings.number_field
[x^2*y^2, x^2, y, 1]


sage: # needs sage.rings.number_field
sage: R.<fx,fy,gx,gy> = QQbar[]
sage: F = (fx*gy - fy*gx)^3; F
-fy^3*gx^3 + 3*fx*fy^2*gx^2*gy + (-3)*fx^2*fy*gx*gy^2 + fx^3*gy^3
sage: F.monomials()
[fy^3*gx^3, fx*fy^2*gx^2*gy, fx^2*fy*gx*gy^2, fx^3*gy^3]
sage: F.coefficients()
[-1, 3, -3, 1]
sage: sum(map(mul, zip(F.coefficients(), F.monomials()))) == F
True

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['fx, fy, gx, gy']; (fx, fy, gx, gy,) = R._first_ngens(4)
>>> F = (fx*gy - fy*gx)**Integer(3); F
-fy^3*gx^3 + 3*fx*fy^2*gx^2*gy + (-3)*fx^2*fy*gx*gy^2 + fx^3*gy^3
>>> F.monomials()
[fy^3*gx^3, fx*fy^2*gx^2*gy, fx^2*fy*gx*gy^2, fx^3*gy^3]
>>> F.coefficients()
[-1, 3, -3, 1]
>>> sum(map(mul, zip(F.coefficients(), F.monomials()))) == F
True
nvariables()[source]

Return the number of variables in this polynomial.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y> = QQbar[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.nvariables()
2
sage: g = f.subs({x: 10}); g
700*y^2 + (-2)*y + 305
sage: g.nvariables()
1

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)
>>> f = Integer(3)*x**Integer(2) - Integer(2)*y + Integer(7)*x**Integer(2)*y**Integer(2) + Integer(5)
>>> f.nvariables()
2
>>> g = f.subs({x: Integer(10)}); g
700*y^2 + (-2)*y + 305
>>> g.nvariables()
1
quo_rem(right)[source]

Return quotient and remainder of self and right.

EXAMPLES:

sage: R.<x,y> = CC[]                                                        # needs sage.rings.real_mpfr
sage: f = y*x^2 + x + 1                                                     # needs sage.rings.real_mpfr
sage: f.quo_rem(x)                                                          # needs sage.libs.singular sage.rings.real_mpfr
(x*y + 1.00000000000000, 1.00000000000000)

sage: R = QQ['a','b']['x','y','z']
sage: p1 = R('a + (1+2*b)*x*y + (3-a^2)*z')
sage: p2 = R('x-1')
sage: p1.quo_rem(p2)                                                        # needs sage.libs.singular
((2*b + 1)*y, (2*b + 1)*y + (-a^2 + 3)*z + a)

sage: R.<x,y> = Qp(5)[]                                                     # needs sage.rings.padics
sage: x.quo_rem(y)                                                          # needs sage.libs.singular sage.rings.padics
Traceback (most recent call last):
...
TypeError: no conversion of this ring to a Singular ring defined

>>> from sage.all import *
>>> R = CC['x, y']; (x, y,) = R._first_ngens(2)# needs sage.rings.real_mpfr
>>> f = y*x**Integer(2) + x + Integer(1)                                                     # needs sage.rings.real_mpfr
>>> f.quo_rem(x)                                                          # needs sage.libs.singular sage.rings.real_mpfr
(x*y + 1.00000000000000, 1.00000000000000)

>>> R = QQ['a','b']['x','y','z']
>>> p1 = R('a + (1+2*b)*x*y + (3-a^2)*z')
>>> p2 = R('x-1')
>>> p1.quo_rem(p2)                                                        # needs sage.libs.singular
((2*b + 1)*y, (2*b + 1)*y + (-a^2 + 3)*z + a)

>>> R = Qp(Integer(5))['x, y']; (x, y,) = R._first_ngens(2)# needs sage.rings.padics
>>> x.quo_rem(y)                                                          # needs sage.libs.singular sage.rings.padics
Traceback (most recent call last):
...
TypeError: no conversion of this ring to a Singular ring defined

ALGORITHM: Use Singular.

reduce(I)[source]

Reduce this polynomial by the polynomials in \(I\).

INPUT:

  • I – list of polynomials or an ideal

EXAMPLES:

sage: # needs sage.rings.number_field
sage: P.<x,y,z> = QQbar[]
sage: f1 = -2 * x^2 + x^3
sage: f2 = -2 * y + x * y
sage: f3 = -x^2 + y^2
sage: F = Ideal([f1, f2, f3])
sage: g = x*y - 3*x*y^2
sage: g.reduce(F)                                                           # needs sage.libs.singular
(-6)*y^2 + 2*y
sage: g.reduce(F.gens())                                                    # needs sage.libs.singular
(-6)*y^2 + 2*y

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> P = QQbar['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> f1 = -Integer(2) * x**Integer(2) + x**Integer(3)
>>> f2 = -Integer(2) * y + x * y
>>> f3 = -x**Integer(2) + y**Integer(2)
>>> F = Ideal([f1, f2, f3])
>>> g = x*y - Integer(3)*x*y**Integer(2)
>>> g.reduce(F)                                                           # needs sage.libs.singular
(-6)*y^2 + 2*y
>>> g.reduce(F.gens())                                                    # needs sage.libs.singular
(-6)*y^2 + 2*y


sage: f = 3*x                                                               # needs sage.rings.number_field
sage: f.reduce([2*x, y])                                                    # needs sage.rings.number_field
0

>>> from sage.all import *
>>> f = Integer(3)*x                                                               # needs sage.rings.number_field
>>> f.reduce([Integer(2)*x, y])                                                    # needs sage.rings.number_field
0


sage: # needs sage.rings.number_field
sage: k.<w> = CyclotomicField(3)
sage: A.<y9,y12,y13,y15> = PolynomialRing(k)
sage: J = [y9 + y12]
sage: f = y9 - y12; f.reduce(J)
-2*y12
sage: f = y13*y15; f.reduce(J)
y13*y15
sage: f = y13*y15 + y9 - y12; f.reduce(J)
y13*y15 - 2*y12

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> k = CyclotomicField(Integer(3), names=('w',)); (w,) = k._first_ngens(1)
>>> A = PolynomialRing(k, names=('y9', 'y12', 'y13', 'y15',)); (y9, y12, y13, y15,) = A._first_ngens(4)
>>> J = [y9 + y12]
>>> f = y9 - y12; f.reduce(J)
-2*y12
>>> f = y13*y15; f.reduce(J)
y13*y15
>>> f = y13*y15 + y9 - y12; f.reduce(J)
y13*y15 - 2*y12

Make sure the remainder returns the correct type, fixing Issue #13903:

sage: R.<y1,y2> = PolynomialRing(Qp(5), 2, order='lex')                     # needs sage.rings.padics
sage: G = [y1^2 + y2^2, y1*y2 + y2^2, y2^3]                                 # needs sage.rings.padics
sage: type((y2^3).reduce(G))                                                # needs sage.rings.padics
<class 'sage.rings.polynomial.multi_polynomial_element.MPolynomial_polydict'>

>>> from sage.all import *
>>> R = PolynomialRing(Qp(Integer(5)), Integer(2), order='lex', names=('y1', 'y2',)); (y1, y2,) = R._first_ngens(2)# needs sage.rings.padics
>>> G = [y1**Integer(2) + y2**Integer(2), y1*y2 + y2**Integer(2), y2**Integer(3)]                                 # needs sage.rings.padics
>>> type((y2**Integer(3)).reduce(G))                                                # needs sage.rings.padics
<class 'sage.rings.polynomial.multi_polynomial_element.MPolynomial_polydict'>
resultant(other, variable=None)[source]

Compute the resultant of self and other with respect to variable.

If a second argument is not provided, the first variable of self.parent() is chosen.

For inexact rings or rings not available in Singular, this computes the determinant of the Sylvester matrix.

INPUT:

  • other – polynomial in self.parent()

  • variable – (optional) variable (of type polynomial) in self.parent()

EXAMPLES:

sage: P.<x,y> = PolynomialRing(QQ, 2)
sage: a = x + y
sage: b = x^3 - y^3
sage: a.resultant(b)                                                        # needs sage.libs.singular
-2*y^3
sage: a.resultant(b, y)                                                     # needs sage.libs.singular
2*x^3

>>> from sage.all import *
>>> P = PolynomialRing(QQ, Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> a = x + y
>>> b = x**Integer(3) - y**Integer(3)
>>> a.resultant(b)                                                        # needs sage.libs.singular
-2*y^3
>>> a.resultant(b, y)                                                     # needs sage.libs.singular
2*x^3
subresultants(other, variable=None)[source]

Return the nonzero subresultant polynomials of self and other.

INPUT:

  • other – a polynomial

OUTPUT: list of polynomials in the same ring as self

EXAMPLES:

sage: # needs sage.libs.singular sage.rings.number_field
sage: R.<x,y> = QQbar[]
sage: p = (y^2 + 6)*(x - 1) - y*(x^2 + 1)
sage: q = (x^2 + 6)*(y - 1) - x*(y^2 + 1)
sage: p.subresultants(q, y)
[2*x^6 + (-22)*x^5 + 102*x^4 + (-274)*x^3 + 488*x^2 + (-552)*x + 288,
 -x^3 - x^2*y + 6*x^2 + 5*x*y + (-11)*x + (-6)*y + 6]
sage: p.subresultants(q, x)
[2*y^6 + (-22)*y^5 + 102*y^4 + (-274)*y^3 + 488*y^2 + (-552)*y + 288,
 x*y^2 + y^3 + (-5)*x*y + (-6)*y^2 + 6*x + 11*y - 6]

>>> from sage.all import *
>>> # needs sage.libs.singular sage.rings.number_field
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)
>>> p = (y**Integer(2) + Integer(6))*(x - Integer(1)) - y*(x**Integer(2) + Integer(1))
>>> q = (x**Integer(2) + Integer(6))*(y - Integer(1)) - x*(y**Integer(2) + Integer(1))
>>> p.subresultants(q, y)
[2*x^6 + (-22)*x^5 + 102*x^4 + (-274)*x^3 + 488*x^2 + (-552)*x + 288,
 -x^3 - x^2*y + 6*x^2 + 5*x*y + (-11)*x + (-6)*y + 6]
>>> p.subresultants(q, x)
[2*y^6 + (-22)*y^5 + 102*y^4 + (-274)*y^3 + 488*y^2 + (-552)*y + 288,
 x*y^2 + y^3 + (-5)*x*y + (-6)*y^2 + 6*x + 11*y - 6]
subs(fixed=None, **kwds)[source]

Fix some given variables in a given multivariate polynomial and return the changed multivariate polynomials. The polynomial itself is not affected. The variable, value pairs for fixing are to be provided as a dictionary of the form {variable: value}.

This is a special case of evaluating the polynomial with some of the variables constants and the others the original variables.

INPUT:

  • fixed – (optional) dictionary of inputs

  • **kwds – named parameters

OUTPUT: new MPolynomial

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y> = QQbar[]
sage: f = x^2 + y + x^2*y^2 + 5
sage: f((5, y))
25*y^2 + y + 30
sage: f.subs({x: 5})
25*y^2 + y + 30

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x**Integer(2) + y + x**Integer(2)*y**Integer(2) + Integer(5)
>>> f((Integer(5), y))
25*y^2 + y + 30
>>> f.subs({x: Integer(5)})
25*y^2 + y + 30
total_degree()[source]

Return the total degree of self, which is the maximum degree of any monomial in self.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y,z> = QQbar[]
sage: f = 2*x*y^3*z^2
sage: f.total_degree()
6
sage: f = 4*x^2*y^2*z^3
sage: f.total_degree()
7
sage: f = 99*x^6*y^3*z^9
sage: f.total_degree()
18
sage: f = x*y^3*z^6 + 3*x^2
sage: f.total_degree()
10
sage: f = z^3 + 8*x^4*y^5*z
sage: f.total_degree()
10
sage: f = z^9 + 10*x^4 + y^8*x^2
sage: f.total_degree()
10

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> f = Integer(2)*x*y**Integer(3)*z**Integer(2)
>>> f.total_degree()
6
>>> f = Integer(4)*x**Integer(2)*y**Integer(2)*z**Integer(3)
>>> f.total_degree()
7
>>> f = Integer(99)*x**Integer(6)*y**Integer(3)*z**Integer(9)
>>> f.total_degree()
18
>>> f = x*y**Integer(3)*z**Integer(6) + Integer(3)*x**Integer(2)
>>> f.total_degree()
10
>>> f = z**Integer(3) + Integer(8)*x**Integer(4)*y**Integer(5)*z
>>> f.total_degree()
10
>>> f = z**Integer(9) + Integer(10)*x**Integer(4) + y**Integer(8)*x**Integer(2)
>>> f.total_degree()
10
univariate_polynomial(R=None)[source]

Return a univariate polynomial associated to this multivariate polynomial.

INPUT:

If this polynomial is not in at most one variable, then a ValueError exception is raised. This is checked using the method is_univariate(). The new Polynomial is over the same base ring as the given MPolynomial.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y> = QQbar[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.univariate_polynomial()
Traceback (most recent call last):
...
TypeError: polynomial must involve at most one variable
sage: g = f.subs({x: 10}); g
700*y^2 + (-2)*y + 305
sage: g.univariate_polynomial()
700*y^2 - 2*y + 305
sage: g.univariate_polynomial(PolynomialRing(QQ, 'z'))
700*z^2 - 2*z + 305

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)
>>> f = Integer(3)*x**Integer(2) - Integer(2)*y + Integer(7)*x**Integer(2)*y**Integer(2) + Integer(5)
>>> f.univariate_polynomial()
Traceback (most recent call last):
...
TypeError: polynomial must involve at most one variable
>>> g = f.subs({x: Integer(10)}); g
700*y^2 + (-2)*y + 305
>>> g.univariate_polynomial()
700*y^2 - 2*y + 305
>>> g.univariate_polynomial(PolynomialRing(QQ, 'z'))
700*z^2 - 2*z + 305
variable(i)[source]

Return the \(i\)-th variable occurring in this polynomial.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y> = QQbar[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.variable(0)
x
sage: f.variable(1)
y

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)
>>> f = Integer(3)*x**Integer(2) - Integer(2)*y + Integer(7)*x**Integer(2)*y**Integer(2) + Integer(5)
>>> f.variable(Integer(0))
x
>>> f.variable(Integer(1))
y
variables()[source]

Return the tuple of variables occurring in this polynomial.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x,y> = QQbar[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.variables()
(x, y)
sage: g = f.subs({x: 10}); g
700*y^2 + (-2)*y + 305
sage: g.variables()
(y,)

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQbar['x, y']; (x, y,) = R._first_ngens(2)
>>> f = Integer(3)*x**Integer(2) - Integer(2)*y + Integer(7)*x**Integer(2)*y**Integer(2) + Integer(5)
>>> f.variables()
(x, y)
>>> g = f.subs({x: Integer(10)}); g
700*y^2 + (-2)*y + 305
>>> g.variables()
(y,)
sage.rings.polynomial.multi_polynomial_element.degree_lowest_rational_function(r, x)[source]

Return the difference of valuations of r with respect to variable x.

INPUT:

  • r – a multivariate rational function

  • x – a multivariate polynomial ring generator

OUTPUT: integer; the difference \(val_x(p) - val_x(q)\) where \(r = p/q\)

Note

This function should be made a method of the FractionFieldElement class.

EXAMPLES:

sage: R1 = PolynomialRing(FiniteField(5), 3, names=["a", "b", "c"])
sage: F = FractionField(R1)
sage: a,b,c = R1.gens()
sage: f = 3*a*b^2*c^3 + 4*a*b*c
sage: g = a^2*b*c^2 + 2*a^2*b^4*c^7

>>> from sage.all import *
>>> R1 = PolynomialRing(FiniteField(Integer(5)), Integer(3), names=["a", "b", "c"])
>>> F = FractionField(R1)
>>> a,b,c = R1.gens()
>>> f = Integer(3)*a*b**Integer(2)*c**Integer(3) + Integer(4)*a*b*c
>>> g = a**Integer(2)*b*c**Integer(2) + Integer(2)*a**Integer(2)*b**Integer(4)*c**Integer(7)

Consider the quotient \(f/g = \frac{4 + 3 bc^{2}}{ac + 2 ab^{3}c^{6}}\) (note the cancellation).

sage: # needs sage.rings.finite_rings
sage: r = f/g; r
(-2*b*c^2 - 1)/(2*a*b^3*c^6 + a*c)
sage: degree_lowest_rational_function(r, a)
-1
sage: degree_lowest_rational_function(r, b)
0
sage: degree_lowest_rational_function(r, c)
-1

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> r = f/g; r
(-2*b*c^2 - 1)/(2*a*b^3*c^6 + a*c)
>>> degree_lowest_rational_function(r, a)
-1
>>> degree_lowest_rational_function(r, b)
0
>>> degree_lowest_rational_function(r, c)
-1