Homogeneous ideals of free algebras#

For twosided ideals and when the base ring is a field, this implementation also provides Groebner bases and ideal containment tests.

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: F
Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F
sage: I
Twosided Ideal (x*y + y*z, x*x + x*y - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> F
Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
>>> I = F*[x*y+y*z,x**Integer(2)+x*y-y*x-y**Integer(2)]*F
>>> I
Twosided Ideal (x*y + y*z, x*x + x*y - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field

One can compute Groebner bases out to a finite degree, can compute normal forms and can test containment in the ideal:

sage: I.groebner_basis(degbound=3)
Twosided Ideal (x*y + y*z,
    x*x - y*x - y*y - y*z,
    y*y*y - y*y*z + y*z*y - y*z*z,
    y*y*x + y*y*z + y*z*x + y*z*z) of Free Associative Unital Algebra
    on 3 generators (x, y, z) over Rational Field
sage: (x*y*z*y*x).normal_form(I)
y*z*z*y*z + y*z*z*z*x + y*z*z*z*z
sage: x*y*z*y*x - (x*y*z*y*x).normal_form(I) in I
True

>>> from sage.all import *
>>> I.groebner_basis(degbound=Integer(3))
Twosided Ideal (x*y + y*z,
    x*x - y*x - y*y - y*z,
    y*y*y - y*y*z + y*z*y - y*z*z,
    y*y*x + y*y*z + y*z*x + y*z*z) of Free Associative Unital Algebra
    on 3 generators (x, y, z) over Rational Field
>>> (x*y*z*y*x).normal_form(I)
y*z*z*y*z + y*z*z*z*x + y*z*z*z*z
>>> x*y*z*y*x - (x*y*z*y*x).normal_form(I) in I
True

AUTHOR:

class sage.algebras.letterplace.letterplace_ideal.LetterplaceIdeal(ring, gens, coerce=True, side='twosided')[source]#

Bases: Ideal_nc

Graded homogeneous ideals in free algebras.

In the two-sided case over a field, one can compute Groebner bases up to a degree bound, normal forms of graded homogeneous elements of the free algebra, and ideal containment.

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F
sage: I
Twosided Ideal (x*y + y*z, x*x + x*y - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: I.groebner_basis(2)
Twosided Ideal (x*y + y*z, x*x - y*x - y*y - y*z) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: I.groebner_basis(4)
Twosided Ideal (x*y + y*z,
    x*x - y*x - y*y - y*z,
    y*y*y - y*y*z + y*z*y - y*z*z,
    y*y*x + y*y*z + y*z*x + y*z*z,
    y*y*z*y - y*y*z*z + y*z*z*y - y*z*z*z,
    y*z*y*y - y*z*y*z + y*z*z*y - y*z*z*z,
    y*y*z*x + y*y*z*z + y*z*z*x + y*z*z*z,
    y*z*y*x + y*z*y*z + y*z*z*x + y*z*z*z) of Free Associative Unital
    Algebra on 3 generators (x, y, z) over Rational Field

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> I = F*[x*y+y*z,x**Integer(2)+x*y-y*x-y**Integer(2)]*F
>>> I
Twosided Ideal (x*y + y*z, x*x + x*y - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
>>> I.groebner_basis(Integer(2))
Twosided Ideal (x*y + y*z, x*x - y*x - y*y - y*z) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
>>> I.groebner_basis(Integer(4))
Twosided Ideal (x*y + y*z,
    x*x - y*x - y*y - y*z,
    y*y*y - y*y*z + y*z*y - y*z*z,
    y*y*x + y*y*z + y*z*x + y*z*z,
    y*y*z*y - y*y*z*z + y*z*z*y - y*z*z*z,
    y*z*y*y - y*z*y*z + y*z*z*y - y*z*z*z,
    y*y*z*x + y*y*z*z + y*z*z*x + y*z*z*z,
    y*z*y*x + y*z*y*z + y*z*z*x + y*z*z*z) of Free Associative Unital
    Algebra on 3 generators (x, y, z) over Rational Field

Groebner bases are cached. If one has computed a Groebner basis out to a high degree then it will also be returned if a Groebner basis with a lower degree bound is requested:

sage: I.groebner_basis(2) is I.groebner_basis(4)
True

>>> from sage.all import *
>>> I.groebner_basis(Integer(2)) is I.groebner_basis(Integer(4))
True

Of course, the normal form of any element has to satisfy the following:

sage: x*y*z*y*x - (x*y*z*y*x).normal_form(I) in I
True

>>> from sage.all import *
>>> x*y*z*y*x - (x*y*z*y*x).normal_form(I) in I
True

Left and right ideals can be constructed, but only twosided ideals provide Groebner bases:

sage: JL = F*[x*y+y*z,x^2+x*y-y*x-y^2]; JL
Left Ideal (x*y + y*z, x*x + x*y - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: JR = [x*y+y*z,x^2+x*y-y*x-y^2]*F; JR
Right Ideal (x*y + y*z, x*x + x*y - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: JR.groebner_basis(2)
Traceback (most recent call last):
...
TypeError: This ideal is not two-sided. We can only compute two-sided Groebner bases
sage: JL.groebner_basis(2)
Traceback (most recent call last):
...
TypeError: This ideal is not two-sided. We can only compute two-sided Groebner bases

>>> from sage.all import *
>>> JL = F*[x*y+y*z,x**Integer(2)+x*y-y*x-y**Integer(2)]; JL
Left Ideal (x*y + y*z, x*x + x*y - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
>>> JR = [x*y+y*z,x**Integer(2)+x*y-y*x-y**Integer(2)]*F; JR
Right Ideal (x*y + y*z, x*x + x*y - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
>>> JR.groebner_basis(Integer(2))
Traceback (most recent call last):
...
TypeError: This ideal is not two-sided. We can only compute two-sided Groebner bases
>>> JL.groebner_basis(Integer(2))
Traceback (most recent call last):
...
TypeError: This ideal is not two-sided. We can only compute two-sided Groebner bases

Also, it is currently not possible to compute a Groebner basis when the base ring is not a field:

sage: FZ.<a,b,c> = FreeAlgebra(ZZ, implementation='letterplace')
sage: J = FZ*[a^3-b^3]*FZ
sage: J.groebner_basis(2)
Traceback (most recent call last):
...
TypeError: Currently, we can only compute Groebner bases if the ring of coefficients is a field

>>> from sage.all import *
>>> FZ = FreeAlgebra(ZZ, implementation='letterplace', names=('a', 'b', 'c',)); (a, b, c,) = FZ._first_ngens(3)
>>> J = FZ*[a**Integer(3)-b**Integer(3)]*FZ
>>> J.groebner_basis(Integer(2))
Traceback (most recent call last):
...
TypeError: Currently, we can only compute Groebner bases if the ring of coefficients is a field

The letterplace implementation of free algebras also provides integral degree weights for the generators, and we can compute Groebner bases for twosided graded homogeneous ideals:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace',degrees=[1,2,3])
sage: I = F*[x*y+z-y*x,x*y*z-x^6+y^3]*F
sage: I.groebner_basis(Infinity)
Twosided Ideal (x*y - y*x + z,
x*x*x*x*x*x - y*x*z - y*y*y + z*z,
x*z*z - y*x*x*z + y*x*z*x + y*y*z + y*z*y + z*x*z + z*y*y - z*z*x,
x*x*x*x*x*z + x*x*x*x*z*x + x*x*x*z*x*x + x*x*z*x*x*x + x*z*x*x*x*x +
y*x*z*y - y*y*x*z + y*z*z + z*x*x*x*x*x - z*z*y,
x*x*x*x*z*y*y + x*x*x*z*y*y*x - x*x*x*z*y*z - x*x*z*y*x*z + x*x*z*y*y*x*x +
x*x*z*y*y*y - x*x*z*y*z*x - x*z*y*x*x*z - x*z*y*x*z*x +
x*z*y*y*x*x*x + 2*x*z*y*y*y*x - 2*x*z*y*y*z - x*z*y*z*x*x -
x*z*y*z*y + y*x*z*x*x*x*x*x - 4*y*x*z*x*x*z - 4*y*x*z*x*z*x +
4*y*x*z*y*x*x*x + 3*y*x*z*y*y*x - 4*y*x*z*y*z + y*y*x*x*x*x*z +
y*y*x*x*x*z*x - 3*y*y*x*x*z*x*x - y*y*x*x*z*y +
5*y*y*x*z*x*x*x + 4*y*y*x*z*y*x - 4*y*y*y*x*x*z +
4*y*y*y*x*z*x + 3*y*y*y*y*z + 4*y*y*y*z*x*x + 6*y*y*y*z*y +
y*y*z*x*x*x*x + y*y*z*x*z + 7*y*y*z*y*x*x + 7*y*y*z*y*y -
7*y*y*z*z*x - y*z*x*x*x*z - y*z*x*x*z*x + 3*y*z*x*z*x*x +
y*z*x*z*y + y*z*y*x*x*x*x - 3*y*z*y*x*z + 7*y*z*y*y*x*x +
3*y*z*y*y*y - 3*y*z*y*z*x - 5*y*z*z*x*x*x - 4*y*z*z*y*x +
4*y*z*z*z - z*y*x*x*x*z - z*y*x*x*z*x - z*y*x*z*x*x -
z*y*x*z*y + z*y*y*x*x*x*x - 3*z*y*y*x*z + 3*z*y*y*y*x*x +
z*y*y*y*y - 3*z*y*y*z*x - z*y*z*x*x*x - 2*z*y*z*y*x +
2*z*y*z*z - z*z*x*x*x*x*x + 4*z*z*x*x*z + 4*z*z*x*z*x -
4*z*z*y*x*x*x - 3*z*z*y*y*x + 4*z*z*y*z + 4*z*z*z*x*x +
2*z*z*z*y)
of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace',degrees=[Integer(1),Integer(2),Integer(3)], names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> I = F*[x*y+z-y*x,x*y*z-x**Integer(6)+y**Integer(3)]*F
>>> I.groebner_basis(Infinity)
Twosided Ideal (x*y - y*x + z,
x*x*x*x*x*x - y*x*z - y*y*y + z*z,
x*z*z - y*x*x*z + y*x*z*x + y*y*z + y*z*y + z*x*z + z*y*y - z*z*x,
x*x*x*x*x*z + x*x*x*x*z*x + x*x*x*z*x*x + x*x*z*x*x*x + x*z*x*x*x*x +
y*x*z*y - y*y*x*z + y*z*z + z*x*x*x*x*x - z*z*y,
x*x*x*x*z*y*y + x*x*x*z*y*y*x - x*x*x*z*y*z - x*x*z*y*x*z + x*x*z*y*y*x*x +
x*x*z*y*y*y - x*x*z*y*z*x - x*z*y*x*x*z - x*z*y*x*z*x +
x*z*y*y*x*x*x + 2*x*z*y*y*y*x - 2*x*z*y*y*z - x*z*y*z*x*x -
x*z*y*z*y + y*x*z*x*x*x*x*x - 4*y*x*z*x*x*z - 4*y*x*z*x*z*x +
4*y*x*z*y*x*x*x + 3*y*x*z*y*y*x - 4*y*x*z*y*z + y*y*x*x*x*x*z +
y*y*x*x*x*z*x - 3*y*y*x*x*z*x*x - y*y*x*x*z*y +
5*y*y*x*z*x*x*x + 4*y*y*x*z*y*x - 4*y*y*y*x*x*z +
4*y*y*y*x*z*x + 3*y*y*y*y*z + 4*y*y*y*z*x*x + 6*y*y*y*z*y +
y*y*z*x*x*x*x + y*y*z*x*z + 7*y*y*z*y*x*x + 7*y*y*z*y*y -
7*y*y*z*z*x - y*z*x*x*x*z - y*z*x*x*z*x + 3*y*z*x*z*x*x +
y*z*x*z*y + y*z*y*x*x*x*x - 3*y*z*y*x*z + 7*y*z*y*y*x*x +
3*y*z*y*y*y - 3*y*z*y*z*x - 5*y*z*z*x*x*x - 4*y*z*z*y*x +
4*y*z*z*z - z*y*x*x*x*z - z*y*x*x*z*x - z*y*x*z*x*x -
z*y*x*z*y + z*y*y*x*x*x*x - 3*z*y*y*x*z + 3*z*y*y*y*x*x +
z*y*y*y*y - 3*z*y*y*z*x - z*y*z*x*x*x - 2*z*y*z*y*x +
2*z*y*z*z - z*z*x*x*x*x*x + 4*z*z*x*x*z + 4*z*z*x*z*x -
4*z*z*y*x*x*x - 3*z*z*y*y*x + 4*z*z*y*z + 4*z*z*z*x*x +
2*z*z*z*y)
of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field

Again, we can compute normal forms:

sage: (z*I.0-I.1).normal_form(I)
0
sage: (z*I.0-x*y*z).normal_form(I)
-y*x*z + z*z

>>> from sage.all import *
>>> (z*I.gen(0)-I.gen(1)).normal_form(I)
0
>>> (z*I.gen(0)-x*y*z).normal_form(I)
-y*x*z + z*z
groebner_basis(degbound=None)[source]#

Twosided Groebner basis with degree bound.

INPUT:

  • degbound (optional integer, or Infinity): If it is provided, a Groebner basis at least out to that degree is returned. By default, the current degree bound of the underlying ring is used.

ASSUMPTIONS:

Currently, we can only compute Groebner bases for twosided ideals, and the ring of coefficients must be a field. A \(TypeError\) is raised if one of these conditions is violated.

Note

  • The result is cached. The same Groebner basis is returned if a smaller degree bound than the known one is requested.

  • If the degree bound Infinity is requested, it is attempted to compute a complete Groebner basis. But we cannot guarantee that the computation will terminate, since not all twosided homogeneous ideals of a free algebra have a finite Groebner basis.

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> I = F*[x*y+y*z,x**Integer(2)+x*y-y*x-y**Integer(2)]*F

Since \(F\) was cached and since its degree bound cannot be decreased, it may happen that, as a side effect of other tests, it already has a degree bound bigger than 3. So, we cannot test against the output of I.groebner_basis():

sage: F.set_degbound(3)
sage: I.groebner_basis()   # not tested
Twosided Ideal (y*y*y - y*y*z + y*z*y - y*z*z, y*y*x + y*y*z + y*z*x + y*z*z, x*y + y*z, x*x - y*x - y*y - y*z) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: I.groebner_basis(4)
Twosided Ideal (x*y + y*z,
    x*x - y*x - y*y - y*z,
    y*y*y - y*y*z + y*z*y - y*z*z,
    y*y*x + y*y*z + y*z*x + y*z*z,
    y*y*z*y - y*y*z*z + y*z*z*y - y*z*z*z,
    y*z*y*y - y*z*y*z + y*z*z*y - y*z*z*z,
    y*y*z*x + y*y*z*z + y*z*z*x + y*z*z*z,
    y*z*y*x + y*z*y*z + y*z*z*x + y*z*z*z) of Free Associative
    Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: I.groebner_basis(2) is I.groebner_basis(4)
True
sage: G = I.groebner_basis(4)
sage: G.groebner_basis(3) is G
True

>>> from sage.all import *
>>> F.set_degbound(Integer(3))
>>> I.groebner_basis()   # not tested
Twosided Ideal (y*y*y - y*y*z + y*z*y - y*z*z, y*y*x + y*y*z + y*z*x + y*z*z, x*y + y*z, x*x - y*x - y*y - y*z) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
>>> I.groebner_basis(Integer(4))
Twosided Ideal (x*y + y*z,
    x*x - y*x - y*y - y*z,
    y*y*y - y*y*z + y*z*y - y*z*z,
    y*y*x + y*y*z + y*z*x + y*z*z,
    y*y*z*y - y*y*z*z + y*z*z*y - y*z*z*z,
    y*z*y*y - y*z*y*z + y*z*z*y - y*z*z*z,
    y*y*z*x + y*y*z*z + y*z*z*x + y*z*z*z,
    y*z*y*x + y*z*y*z + y*z*z*x + y*z*z*z) of Free Associative
    Unital Algebra on 3 generators (x, y, z) over Rational Field
>>> I.groebner_basis(Integer(2)) is I.groebner_basis(Integer(4))
True
>>> G = I.groebner_basis(Integer(4))
>>> G.groebner_basis(Integer(3)) is G
True

If a finite complete Groebner basis exists, we can compute it as follows:

sage: I = F*[x*y-y*x,x*z-z*x,y*z-z*y,x^2*y-z^3,x*y^2+z*x^2]*F
sage: I.groebner_basis(Infinity)
Twosided Ideal (-y*z + z*y,
    -x*z + z*x,
    -x*y + y*x,
    x*x*z + x*y*y,
    x*x*y - z*z*z,
    x*x*x*z + y*z*z*z,
    x*z*z*z*z + y*y*z*z*z) of Free Associative Unital Algebra
    on 3 generators (x, y, z) over Rational Field

>>> from sage.all import *
>>> I = F*[x*y-y*x,x*z-z*x,y*z-z*y,x**Integer(2)*y-z**Integer(3),x*y**Integer(2)+z*x**Integer(2)]*F
>>> I.groebner_basis(Infinity)
Twosided Ideal (-y*z + z*y,
    -x*z + z*x,
    -x*y + y*x,
    x*x*z + x*y*y,
    x*x*y - z*z*z,
    x*x*x*z + y*z*z*z,
    x*z*z*z*z + y*y*z*z*z) of Free Associative Unital Algebra
    on 3 generators (x, y, z) over Rational Field

Since the commutators of the generators are contained in the ideal, we can verify the above result by a computation in a polynomial ring in negative lexicographic order:

sage: P.<c,b,a> = PolynomialRing(QQ,order='neglex')
sage: J = P*[a^2*b-c^3,a*b^2+c*a^2]
sage: J.groebner_basis()
[b*a^2 - c^3, b^2*a + c*a^2, c*a^3 + c^3*b, c^3*b^2 + c^4*a]

>>> from sage.all import *
>>> P = PolynomialRing(QQ,order='neglex', names=('c', 'b', 'a',)); (c, b, a,) = P._first_ngens(3)
>>> J = P*[a**Integer(2)*b-c**Integer(3),a*b**Integer(2)+c*a**Integer(2)]
>>> J.groebner_basis()
[b*a^2 - c^3, b^2*a + c*a^2, c*a^3 + c^3*b, c^3*b^2 + c^4*a]

Apparently, the results are compatible, by sending \(a\) to \(x\), \(b\) to \(y\) and \(c\) to \(z\).

reduce(G)[source]#

Reduction of this ideal by another ideal, or normal form of an algebra element with respect to this ideal.

INPUT:

  • G: A list or tuple of elements, an ideal, the ambient algebra, or a single element.

OUTPUT:

  • The normal form of G with respect to this ideal, if G is an element of the algebra.

  • The reduction of this ideal by the elements resp. generators of G, if G is a list, tuple or ideal.

  • The zero ideal, if G is the algebra containing this ideal.

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F
sage: I.reduce(F)
Twosided Ideal (0) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: I.reduce(x^3)
-y*z*x - y*z*y - y*z*z
sage: I.reduce([x*y])
Twosided Ideal (y*z, x*x - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage: I.reduce(F*[x^2+x*y,y^2+y*z]*F)
Twosided Ideal (x*y + y*z, -y*x + y*z) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field

>>> from sage.all import *
>>> F = FreeAlgebra(QQ, implementation='letterplace', names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> I = F*[x*y+y*z,x**Integer(2)+x*y-y*x-y**Integer(2)]*F
>>> I.reduce(F)
Twosided Ideal (0) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
>>> I.reduce(x**Integer(3))
-y*z*x - y*z*y - y*z*z
>>> I.reduce([x*y])
Twosided Ideal (y*z, x*x - y*x - y*y) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
>>> I.reduce(F*[x**Integer(2)+x*y,y**Integer(2)+y*z]*F)
Twosided Ideal (x*y + y*z, -y*x + y*z) of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field
sage.algebras.letterplace.letterplace_ideal.poly_reduce(ring=None, interruptible=True, attributes=None, *args)[source]#

This function is an automatically generated C wrapper around the Singular function ‘NF’.

This wrapper takes care of converting Sage datatypes to Singular datatypes and vice versa. In addition to whatever parameters the underlying Singular function accepts when called, this function also accepts the following keyword parameters:

INPUT:

  • args – a list of arguments

  • ring – a multivariate polynomial ring

  • interruptible – if True pressing Ctrl + C during the execution of this function will interrupt the computation (default: True)

  • attributes – a dictionary of optional Singular attributes assigned to Singular objects (default: None)

If ring is not specified, it is guessed from the given arguments. If this is not possible, then a dummy ring, univariate polynomial ring over QQ, is used.

EXAMPLES:

sage: groebner = sage.libs.singular.function_factory.ff.groebner
sage: P.<x, y> = PolynomialRing(QQ)
sage: I = P.ideal(x^2-y, y+x)
sage: groebner(I)
[x + y, y^2 - y]
sage: triangL = sage.libs.singular.function_factory.ff.triang__lib.triangL
sage: P.<x1, x2> = PolynomialRing(QQ, 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(Ideal(f1,f2).groebner_basis()[::-1])
sage: triangL(I, attributes={I:{'isSB':1}})
[[x2^4 + 4*x2^3 - 6*x2^2 - 20*x2 + 5, 8*x1 - x2^3 + x2^2 + 13*x2 - 5],
 [x2, x1^2],
 [x2, x1^2],
 [x2, x1^2]]

>>> from sage.all import *
>>> groebner = sage.libs.singular.function_factory.ff.groebner
>>> P = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> I = P.ideal(x**Integer(2)-y, y+x)
>>> groebner(I)
[x + y, y^2 - y]
>>> triangL = sage.libs.singular.function_factory.ff.triang__lib.triangL
>>> P = PolynomialRing(QQ, order='lex', names=('x1', 'x2',)); (x1, x2,) = P._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(Ideal(f1,f2).groebner_basis()[::-Integer(1)])
>>> triangL(I, attributes={I:{'isSB':Integer(1)}})
[[x2^4 + 4*x2^3 - 6*x2^2 - 20*x2 + 5, 8*x1 - x2^3 + x2^2 + 13*x2 - 5],
 [x2, x1^2],
 [x2, x1^2],
 [x2, x1^2]]

The Singular documentation for ‘NF’ is given below.

5.1.131 reduce
--------------

`*Syntax:*'
     `reduce (' poly_expression`,' ideal_expression `)'
     `reduce (' poly_expression`,' ideal_expression`,' int_expression
     `)'
     `reduce (' poly_expression`,' poly_expression`,' ideal_expression
     `)'
     `reduce (' vector_expression`,' ideal_expression `)'
     `reduce (' vector_expression`,' ideal_expression`,' int_expression
     `)'
     `reduce (' vector_expression`,' module_expression `)'
     `reduce (' vector_expression`,' module_expression`,'
     int_expression `)'
     `reduce (' vector_expression`,' poly_expression`,'
     module_expression `)'
     `reduce (' ideal_expression`,' ideal_expression `)'
     `reduce (' ideal_expression`,' ideal_expression`,' int_expression
     `)'
     `reduce (' ideal_expression`,' matrix_expression`,'
     ideal_expression `)'
     `reduce (' module_expression`,' ideal_expression `)'
     `reduce (' module_expression`,' ideal_expression`,' int_expression
     `)'
     `reduce (' module_expression`,' module_expression `)'
     `reduce (' module_expression`,' module_expression`,'
     int_expression `)'
     `reduce (' module_expression`,' matrix_expression`,'
     module_expression `)'
     `reduce (' poly/vector/ideal/module`,' ideal/module`,' int`,'
     intvec `)'
     `reduce (' ideal`,' matrix`,' ideal`,' int `)'
     `reduce (' poly`,' poly`,' ideal`,' int `)'
     `reduce (' poly`,' poly`,' ideal`,' int`,' intvec `)'

`*Type:*'
     the type of the first argument

`*Purpose:*'
     reduces a polynomial, vector, ideal  or module to its normal form
     with respect to an ideal or module represented by a standard basis.
     Returns 0 if and only if the polynomial (resp. vector, ideal,
     module) is an element (resp. subideal, submodule) of the ideal
     (resp. module).  The result may have no meaning if the second
     argument is not a standard basis.
     The third (optional) argument of type int modifies the behavior:
        * 0 default

        * 1 consider only the leading term and do no tail reduction.

        * 2 tail reduction:n the local/mixed ordering case: reduce also
          with bad ecart

        * 4 reduce without division, return possibly a non-zero
          constant multiple of the remainder

     If a second argument `u' of type poly or matrix is given, the
     first argument `p' is replaced by `p/u'.  This works only for zero
     dimensional ideals (resp. modules) in the third argument and
     gives, even in a local ring, a reduced normal form which is the
     projection to the quotient by the ideal (resp. module).  One may
     give a degree bound in the fourth argument with respect to a
     weight vector in the fifth argument in order have a finite
     computation.  If some of the weights are zero, the procedure may
     not terminate!

`*Note_*'
     The commands `reduce' and `NF' are synonymous.

`*Example:*'
            ring r1 = 0,(z,y,x),ds;
            poly s1=2x5y+7x2y4+3x2yz3;
            poly s2=1x2y2z2+3z8;
            poly s3=4xy5+2x2y2z3+11x10;
            ideal i=s1,s2,s3;
            ideal j=std(i);
            reduce(3z3yx2+7y4x2+yx5+z12y2x2,j);
          ==> -yx5+2401/81y14x2+2744/81y11x5+392/27y8x8+224/81y5x11+16/81y2x14
            reduce(3z3yx2+7y4x2+yx5+z12y2x2,j,1);
          ==> -yx5+z12y2x2
            // 4 arguments:
            ring rs=0,x,ds;
            // normalform of 1/(1+x) w.r.t. (x3) up to degree 5
            reduce(poly(1),1+x,ideal(x3),5);
          ==> // ** _ is no standard basis
          ==> 1-x+x2

* Menu:

See
* division::
* ideal::
* module::
* poly operations::
* std::
* vector::
sage.algebras.letterplace.letterplace_ideal.singular_twostd(ring=None, interruptible=True, attributes=None, *args)[source]#

This function is an automatically generated C wrapper around the Singular function ‘twostd’.

This wrapper takes care of converting Sage datatypes to Singular datatypes and vice versa. In addition to whatever parameters the underlying Singular function accepts when called, this function also accepts the following keyword parameters:

INPUT:

  • args – a list of arguments

  • ring – a multivariate polynomial ring

  • interruptible – if True pressing Ctrl + C during the execution of this function will interrupt the computation (default: True)

  • attributes – a dictionary of optional Singular attributes assigned to Singular objects (default: None)

If ring is not specified, it is guessed from the given arguments. If this is not possible, then a dummy ring, univariate polynomial ring over QQ, is used.

EXAMPLES:

sage: groebner = sage.libs.singular.function_factory.ff.groebner
sage: P.<x, y> = PolynomialRing(QQ)
sage: I = P.ideal(x^2-y, y+x)
sage: groebner(I)
[x + y, y^2 - y]
sage: triangL = sage.libs.singular.function_factory.ff.triang__lib.triangL
sage: P.<x1, x2> = PolynomialRing(QQ, 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(Ideal(f1,f2).groebner_basis()[::-1])
sage: triangL(I, attributes={I:{'isSB':1}})
[[x2^4 + 4*x2^3 - 6*x2^2 - 20*x2 + 5, 8*x1 - x2^3 + x2^2 + 13*x2 - 5],
 [x2, x1^2],
 [x2, x1^2],
 [x2, x1^2]]

>>> from sage.all import *
>>> groebner = sage.libs.singular.function_factory.ff.groebner
>>> P = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> I = P.ideal(x**Integer(2)-y, y+x)
>>> groebner(I)
[x + y, y^2 - y]
>>> triangL = sage.libs.singular.function_factory.ff.triang__lib.triangL
>>> P = PolynomialRing(QQ, order='lex', names=('x1', 'x2',)); (x1, x2,) = P._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(Ideal(f1,f2).groebner_basis()[::-Integer(1)])
>>> triangL(I, attributes={I:{'isSB':Integer(1)}})
[[x2^4 + 4*x2^3 - 6*x2^2 - 20*x2 + 5, 8*x1 - x2^3 + x2^2 + 13*x2 - 5],
 [x2, x1^2],
 [x2, x1^2],
 [x2, x1^2]]

The Singular documentation for ‘twostd’ is given below.

5.1.155 system
--------------

`*Syntax:*'
     `system (' string_expression `)'
     `system (' string_expression`,' expression `)'

`*Type:*'
     depends on the desired function, may be none

`*Purpose:*'
     interface to internal data and the operating system. The
     string_expression determines the command to execute. Some commands
     require an additional argument (second form) where the type of the
     argument depends on the command. See below for a list of all
     possible commands.

`*Note_*'
     Not all functions work on every platform.

`*Functions:*'

    `system("alarm",' int `)'
          abort the Singular process after computing for that many
          seconds (system+user cpu time).

    `system("absFact",' poly `)'
          absolute factorization of the polynomial (from a polynomial
          ring over a transzedental extension) Returns a list of the
          ideal of the factors, intvec of multiplicities, ideal of
          minimal polynomials and the number of factors.

    `system("blackbox")'
          list all blackbox data types.

    `system("browsers");'
          returns a string about available help browsers.  *Note The
          online help system::.

    `system("bracket",' poly, poly `)'
          returns the Lie bracket [p,q].

    `system("complexNearZero",' number_expression `)'
          checks for a small value for floating point numbers

    `system("contributors")'
          returns names of people who contributed to the SINGULAR
          kernel as string.

    `system("content",p)'
          returns p/content(p) for poly/vector

    `system("cpu")'
          returns the number of cpus as int (for creating multiple
          threads/processes).  (see `system("--cpus")').

    `system("denom_list")'
          returns the list of denominators (number) which occurred in
          the latest std computationi(s).  Is reset to the empty list
          at ring changes or by this system call.

    `system("eigenvals",' matrix `)'
          returns the list of the eigenvalues of the matrix (as ideal,
          intvec).  (see `system("hessenberg")').

    `system("env",' ring `)'
          returns the enveloping algebra (i.e. R tensor R^opp) See
          `system("opp")'.

    `system("executable",' string `)'
          returns the path of the command given as argument or the
          empty string (for: not found) See `system("Singular")'.  See
          `system("getenv","PATH")'.

    `system("getenv",' string_expression`)'
          returns the value of the shell environment variable given as
          the second argument. The return type is string.

    `system("getPrecDigits")'
          returns the precision for floating point numbers

    `system("gmsnf",' ideal, ideal, matrix,int, int `)'
          Gauss-Manin system: for gmspoly.lib, gmssing.lib

    `system("HC")'
          returns the degree of the "highest corner" from the last std
          computation (or 0).

    `system("hessenberg",' matrix `)'
          returns the Hessenberg matrix (via QR algorithm).

    `system("install",' s1, s2, p3, i4 `)'
          install a new method p3 for s2 for the newstruct type s1.  s2
          must be a reserved operator with i4 operands (i4 may be
          1,2,3; use 4 for more than 3 or a varying number of arguments)
          See *Note Commands for user defined types::.

    `system("LLL",' B `)'
          B must be a matrix or an intmat.  Interface to NTLs LLL
          (Exact Arithmetic Variant over ZZ).  Returns the same type as
          the input.
          B is an m x n matrix, viewed as m rows of n-vectors.  m may
          be less than, equal to, or greater than n, and the rows need
          not be linearly independent.  B is transformed into an
          LLL-reduced basis.  The first m-rank(B) rows of B are zero.
          More specifically, elementary row transformations are
          performed on B so that the non-zero rows of new-B form an
          LLL-reduced basis for the lattice spanned by the rows of
          old-B.

    `system("nblocks")' or `system("nblocks",' ring_name `)'
          returns the number of blocks of the given ring, or of the
          current basering, if no second argument is given. The return
          type is int.

    `system("nc_hilb",' ideal, int, [,...] `)'
          internal support for ncHilb.lib, return nothing

    `system("neworder",' ideal `)'
          string of the ring variables in an heurically good order for
          `char_series'

    `system("newstruct")'
          list all newstruct data types.

    `system("opp",' ring `)'
          returns the opposite ring.

    `system("oppose",' ring R, poly p `)'
          returns the opposite polynomial of p from R.

    `system("pcvLAddL",' list, list `)'
          `system("pcvPMulL",' poly, list `)' 
          `system("pcvMinDeg",' poly `)' 
          `system("pcvP2CV",' list, int, int `)' 
          `system("pcvCV2P",' list, int, int `)' 
          `system("pcvDim",' int, int `)' 
          `system("pcvBasis",' int, int `)' internal for mondromy.lib

    `system("pid")'
          returns the process number as int (for creating unique names).

    `system("random")' or `system("random",' int `)'
          returns or sets the seed of the random generator.

    `system("reduce_bound",' poly, ideal, int `)'
          or `system("reduce_bound",' ideal, ideal, int `)'
          or `system("reduce_bound",' vector, module, int `)'
          or `system("reduce_bound",' module, module, int `)' returns
          the normalform of the first argument wrt. the second up to
          the given degree bound (wrt. total degree)

    `system("reserve",' int `)'
          reserve a port and listen with the given backlog.  (see
          `system("reservedLink")').

    `system("reservedLink")'
          accept a connect at the reserved port and return a
          (write-only) link to it.  (see `system("reserve")').

    `system("rref",' matrix ` )'
          return a reduced row echelon form of the constant matrix M
          (see `system("rref")').

    `system("semaphore",' string, int `)'
          operations for semaphores: string may be `"init"', `"exists"',
          `"acquire"', `"try_acquire"', `"release"', `"get_value"', and
          int is the number of the semaphore.  Returns -2 for wrong
          command, -1 for error or the result of the command.

    `system("semic",' list, list `)'
          or `system("semic",' list, list, int `)' computes from list
          of spectrum numbers and list of spectrum numbers the
          semicontinuity index (qh, if 3rd argument is 1).

    `system("setenv",'string_expression, string_expression`)'
          sets the shell environment variable given as the second
          argument to the value given as the third argument. Returns
          the third argument. Might not be available on all platforms.

    `system("sh"', string_expression `)'
          shell escape, returns the return code of the shell as int.
          The string is sent literally to the shell.

    `system("shrinktest",' poly, i2 `)'
          internal for shift algebra (with i2 variables): shrink the
          poly

    `system("Singular")'
          returns the absolute (path) name of the running SINGULAR as
          string.

    `system("SingularBin")'
          returns the absolute path name of directory of the running
          SINGULAR as string (ending in /)

    `system("SingularLib")'
          returns the colon separated library search path name as
          string.

    `system("spadd",' list, list `)'
          or `system("spadd",' list, list, int `)' computes from list
          of spectrum numbers and list of spectrum numbers the sum of
          the lists.

    `system("spectrum",' poly `)'
          or  `system("spectrum",' poly, int `)' 

    `system("spmul",' list, int `)'
          or `system("spmul",' list, list, int `)' computes from list
          of spectrum numbers the multiple of it.

    `system("std_syz",' module, int `)'
          compute a partial groebner base of a module, stop after the
          given column

    `system("tensorModuleMult",' int, module `)'
          internal for sheafcoh.lib (see id_TensorModuleMult)

    `system("twostd",' ideal `)'
          returns the two-sided standard basis of the two-sided ideal.

    `system("uname")'
          returns a string identifying the architecture for which
          SINGULAR was compiled.

    `system("verifyGB",' ideal_expression/module_expression `)'
          checks, if an ideal/module is a Groebner base

    `system("version")'
          returns the version number of SINGULAR as int.  (Version
          a-b-c-d returns a*1000+b*100+c*10+d)

    `system("with")'
          without an argument: returns a string describing the current
          version of SINGULAR, its build options, the used path names
          and other configurations
          with a string argument: test for that feature and return an
          int.

    `system("--cpus")'
          returns the number of available cpu cores as int (for using
          multiple cores).  (see `system("cpu")').

    `system("'-`")'
          prints the values of all options.

    `system("'-long_option_name`")'
          returns the value of the (command-line) option
          long_option_name. The type of the returned value is either
          string or int.  *Note Command line options::, for more info.

    `system("'-long_option_name`",' expression`)'
          sets the value of the (command-line) option long_option_name
          to the value given by the expression. Type of the expression
          must be string, or int. *Note Command line options::, for
          more info. Among others, this can be used for setting the
          seed of the random number generator, the used help browser,
          the minimal display time, or the timer resolution.

`*Example:*'
          // a listing of the current directory:
          system("sh","ls");
          // execute a shell, return to SINGULAR with exit:
          system("sh","sh");
          string unique_name="/tmp/xx"+string(system("pid"));
          unique_name;
          ==> /tmp/xx4711
          system("uname")
          ==> ix86-Linux
          system("getenv","PATH");
          ==> /bin:/usr/bin:/usr/local/bin
          system("Singular");
          ==> /usr/local/bin/Singular
          // report value of all options
          system("--");
          ==> // --batch           0
          ==> // --execute
          ==> // --sdb             0
          ==> // --echo            1
          ==> // --profile         0
          ==> // --quiet           1
          ==> // --sort            0
          ==> // --random          12345678
          ==> // --no-tty          1
          ==> // --user-option
          ==> // --allow-net       0
          ==> // --browser
          ==> // --cntrlc
          ==> // --emacs           0
          ==> // --log
          ==> // --no-stdlib       0
          ==> // --no-rc           1
          ==> // --no-warn         0
          ==> // --no-out          0
          ==> // --no-shell        0
          ==> // --min-time        "0.5"
          ==> // --cpus            4
          ==> // --threads         4
          ==> // --flint-threads   1
          ==> // --MPport
          ==> // --MPhost
          ==> // --link
          ==> // --ticks-per-sec   1
          // set minimal display time to 0.02 seconds
          system("--min-time", "0.02");
          // set timer resolution to 0.01 seconds
          system("--ticks-per-sec", 100);
          // re-seed random number generator
          system("--random", 12345678);
          // allow your web browser to access HTML pages from the net
          system("--allow-net", 1);
          // and set help browser to firefox
          system("--browser", "firefox");
          ==> // ** No help browser 'firefox' available.
          ==> // ** Setting help browser to 'dummy'.