Weighted homogeneous elements of free algebras, in letterplace implementation.

AUTHOR:

class sage.algebras.letterplace.free_algebra_element_letterplace.FreeAlgebraElement_letterplace

Bases: sage.structure.element.AlgebraElement

Weighted homogeneous elements of a free associative unital algebra (letterplace implementation)

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: x+y
x + y
sage: x*y !=y*x
True
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F
sage: (y^3).reduce(I)
y*y*y
sage: (y^3).normal_form(I)
y*y*z - y*z*y + y*z*z

Here is an example with nontrivial degree weights:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[2,1,3])
sage: I = F*[x*y-y*x, x^2+2*y*z, (x*y)^2-z^2]*F
sage: x.degree()
2
sage: y.degree()
1
sage: z.degree()
3
sage: (x*y)^3
x*y*x*y*x*y
sage: ((x*y)^3).normal_form(I)
z*z*y*x
sage: ((x*y)^3).degree()
9
degree()

Return the degree of this element.

NOTE:

Generators may have a positive integral degree weight. All elements must be weighted homogeneous.

EXAMPLE:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: ((x+y+z)^3).degree()
3
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[2,1,3])
sage: ((x*y+z)^3).degree()
9
lc()

The leading coefficient of this free algebra element, as element of the base ring.

EXAMPLE:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: ((2*x+3*y-4*z)^2*(5*y+6*z)).lc()
20
sage: ((2*x+3*y-4*z)^2*(5*y+6*z)).lc().parent() is F.base()
True
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[2,1,3])
sage: ((2*x*y+z)^2).lc()
4
letterplace_polynomial()

Return the commutative polynomial that is used internally to represent this free algebra element.

EXAMPLE:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: ((x+y-z)^2).letterplace_polynomial()
x*x_1 + x*y_1 - x*z_1 + y*x_1 + y*y_1 - y*z_1 - z*x_1 - z*y_1 + z*z_1

If degree weights are used, the letterplace polynomial is homogenized by slack variables:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[2,1,3])
sage: ((x*y+z)^2).letterplace_polynomial()
x*x__1*y_2*x_3*x__4*y_5 + x*x__1*y_2*z_3*x__4*x__5 + z*x__1*x__2*x_3*x__4*y_5 + z*x__1*x__2*z_3*x__4*x__5
lm()

The leading monomial of this free algebra element.

EXAMPLE:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: ((2*x+3*y-4*z)^2*(5*y+6*z)).lm()
x*x*y
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[2,1,3])
sage: ((2*x*y+z)^2).lm()
x*y*x*y
lm_divides(p)

Tell whether or not the leading monomial of self devides the leading monomial of another element.

NOTE:

A free algebra element p divides another one q if there are free algebra elements s and t such that spt = q.

EXAMPLE:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[2,1,3])
sage: ((2*x*y+z)^2*z).lm()
x*y*x*y*z
sage: (y*x*y-y^4).lm()
y*x*y
sage: (y*x*y-y^4).lm_divides((2*x*y+z)^2*z)
True
lt()

The leading term (monomial times coefficient) of this free algebra element.

EXAMPLE:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: ((2*x+3*y-4*z)^2*(5*y+6*z)).lt()
20*x*x*y
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[2,1,3])
sage: ((2*x*y+z)^2).lt()
4*x*y*x*y
normal_form(I)

Return the normal form of this element with respect to a twosided weighted homogeneous ideal.

INPUT:

A twosided homogeneous ideal I of the parent F of this element, x.

OUTPUT:

The normal form of x wrt. I.

NOTE:

The normal form is computed by reduction with respect to a Groebnerbasis of I with degree bound deg(x).

EXAMPLE:

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: (x^5).normal_form(I)
-y*z*z*z*x - y*z*z*z*y - y*z*z*z*z

We verify two basic properties of normal forms: The difference of an element and its normal form is contained in the ideal, and if two elements of the free algebra differ by an element of the ideal then they have the same normal form:

sage: x^5 - (x^5).normal_form(I) in I
True
sage: (x^5+x*I.0*y*z-3*z^2*I.1*y).normal_form(I) == (x^5).normal_form(I)
True

Here is an example with non-trivial degree weights:

sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace', degrees=[1,2,3])
sage: I = F*[x*y-y*x+z, y^2+2*x*z, (x*y)^2-z^2]*F
sage: ((x*y)^3).normal_form(I)
z*z*y*x - z*z*z
sage: (x*y)^3-((x*y)^3).normal_form(I) in I
True
sage: ((x*y)^3+2*z*I.0*z+y*I.1*z-x*I.2*y).normal_form(I) == ((x*y)^3).normal_form(I)
True
reduce(G)

Reduce this element by a list of elements or by a twosided weighted homogeneous ideal.

INPUT:

Either a list or tuple of weighted homogeneous elements of the free algebra, or an ideal of the free algebra, or an ideal in the commutative polynomial ring that is currently used to implement the multiplication in the free algebra.

OUTPUT:

The twosided reduction of this element by the argument.

NOTE:

This may not be the normal form of this element, unless the argument is a twosided Groebner basis up to the degree of this element.

EXAMPLE:

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: p = y^2*z*y^2+y*z*y*z*y

We compute the letterplace version of the Groebneer basis of I with degree bound 4:

sage: G = F._reductor_(I.groebner_basis(4).gens(),4)
sage: G.ring() is F.current_ring()
True

Since the element p is of degree 5, it is no surrprise that its reductions with respect to the original generators of I (of degree 2), or with respect to G (Groebner basis with degree bound 4), or with respect to the Groebner basis with degree bound 5 (which yields its normal form) are pairwise different:

sage: p.reduce(I)
y*y*z*y*y + y*z*y*z*y
sage: p.reduce(G)
y*y*z*z*y + y*z*y*z*y - y*z*z*y*y + y*z*z*z*y
sage: p.normal_form(I)
y*y*z*z*z + y*z*y*z*z - y*z*z*y*z + y*z*z*z*z
sage: p.reduce(I) != p.reduce(G) != p.normal_form(I) != p.reduce(I)
True
sage.algebras.letterplace.free_algebra_element_letterplace.poly_reduce(ring=None, interruptible=True, attributes=None, *args)

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)

EXAMPLE:

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]]

The Singular documentation for ‘NF’ is given below.

Singular documentation not found
sage.algebras.letterplace.free_algebra_element_letterplace.singular_system(ring=None, interruptible=True, attributes=None, *args)

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

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)

EXAMPLE:

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]]

The Singular documentation for ‘system’ is given below.

Singular documentation not found