Noncommutative polynomials via libSINGULAR/Plural#

This module provides specialized and optimized implementations for noncommutative multivariate polynomials over many coefficient rings, via the shared library interface to SINGULAR. In particular, the following coefficient rings are supported by this implementation:

  • the rational numbers \(\QQ\), and

  • finite fields \(\GF{p}\) for \(p\) prime

AUTHORS:

The PLURAL wrapper is due to

  • Burcin Erocal (2008-11 and 2010-07): initial implementation and concept

  • Michael Brickenstein (2008-11 and 2010-07): initial implementation and concept

  • Oleksandr Motsak (2010-07): complete overall noncommutative functionality and first release

  • Alexander Dreyer (2010-07): noncommutative ring functionality and documentation

  • Simon King (2011-09): left and two-sided ideals; normal forms; pickling; documentation

The underlying libSINGULAR interface was implemented by

  • Martin Albrecht (2007-01): initial implementation

  • Joel Mohler (2008-01): misc improvements, polishing

  • Martin Albrecht (2008-08): added \(\QQ(a)\) and \(\ZZ\) support

  • Simon King (2009-04): improved coercion

  • Martin Albrecht (2009-05): added \(\ZZ/n\ZZ\) support, refactoring

  • Martin Albrecht (2009-06): refactored the code to allow better re-use

Todo

extend functionality towards those of libSINGULARs commutative part

EXAMPLES:

We show how to construct various noncommutative polynomial rings:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P.<x,y,z> = A.g_algebra(relations={y*x:-x*y}, order = 'lex')

sage: P
Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -x*y}

sage: y*x + 1/2
-x*y + 1/2

sage: A.<x,y,z> = FreeAlgebra(GF(17), 3)
sage: P.<x,y,z> = A.g_algebra(relations={y*x:-x*y}, order = 'lex')
sage: P
Noncommutative Multivariate Polynomial Ring in x, y, z over Finite Field of size 17, nc-relations: {y*x: -x*y}

sage: y*x + 7
-x*y + 7

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y}, order = 'lex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)

>>> P
Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -x*y}

>>> y*x + Integer(1)/Integer(2)
-x*y + 1/2

>>> A = FreeAlgebra(GF(Integer(17)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y}, order = 'lex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> P
Noncommutative Multivariate Polynomial Ring in x, y, z over Finite Field of size 17, nc-relations: {y*x: -x*y}

>>> y*x + Integer(7)
-x*y + 7

Raw use of this class; this is not the intended use!

sage: from sage.matrix.constructor import Matrix
sage: c = Matrix(3)
sage: c[0,1] = -2
sage: c[0,2] = 1
sage: c[1,2] = 1

sage: d = Matrix(3)
sage: d[0, 1] = 17
sage: P = QQ['x','y','z']
sage: c = c.change_ring(P)
sage: d = d.change_ring(P)

sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural
sage: R.<x,y,z> = NCPolynomialRing_plural(QQ, c = c, d = d, order=TermOrder('lex',3),category=Algebras(QQ))
sage: R
Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -2*x*y + 17}

sage: R.term_order()
Lexicographic term order

sage: a,b,c = R.gens()
sage: f = 57 * a^2*b + 43 * c + 1; f
57*x^2*y + 43*z + 1

>>> from sage.all import *
>>> from sage.matrix.constructor import Matrix
>>> c = Matrix(Integer(3))
>>> c[Integer(0),Integer(1)] = -Integer(2)
>>> c[Integer(0),Integer(2)] = Integer(1)
>>> c[Integer(1),Integer(2)] = Integer(1)

>>> d = Matrix(Integer(3))
>>> d[Integer(0), Integer(1)] = Integer(17)
>>> P = QQ['x','y','z']
>>> c = c.change_ring(P)
>>> d = d.change_ring(P)

>>> from sage.rings.polynomial.plural import NCPolynomialRing_plural
>>> R = NCPolynomialRing_plural(QQ, c = c, d = d, order=TermOrder('lex',Integer(3)),category=Algebras(QQ), names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> R
Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -2*x*y + 17}

>>> R.term_order()
Lexicographic term order

>>> a,b,c = R.gens()
>>> f = Integer(57) * a**Integer(2)*b + Integer(43) * c + Integer(1); f
57*x^2*y + 43*z + 1
sage.rings.polynomial.plural.ExteriorAlgebra(base_ring, names, order='degrevlex')[source]#

Return the exterior algebra on some generators.

This is also known as a Grassmann algebra. This is a finite dimensional algebra, where all generators anti-commute.

See Wikipedia article Exterior algebra

INPUT:

  • base_ring – the ground ring

  • names – a list of variable names

EXAMPLES:

sage: from sage.rings.polynomial.plural import ExteriorAlgebra
sage: E = ExteriorAlgebra(QQ, ['x', 'y', 'z']) ; E #random
Quotient of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: -x*z, z*y: -y*z, y*x: -x*y} by the ideal (z^2, y^2, x^2)
sage: sorted(E.cover().domain().relations().items(), key=str)
[(y*x, -x*y), (z*x, -x*z), (z*y, -y*z)]
sage: sorted(E.cover().kernel().gens(),key=str)
[x^2, y^2, z^2]
sage: E.inject_variables()
Defining xbar, ybar, zbar
sage: x,y,z = (xbar,ybar,zbar)
sage: y*x
-x*y
sage: all(v^2==0 for v in E.gens())
True
sage: E.one()
1

>>> from sage.all import *
>>> from sage.rings.polynomial.plural import ExteriorAlgebra
>>> E = ExteriorAlgebra(QQ, ['x', 'y', 'z']) ; E #random
Quotient of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: -x*z, z*y: -y*z, y*x: -x*y} by the ideal (z^2, y^2, x^2)
>>> sorted(E.cover().domain().relations().items(), key=str)
[(y*x, -x*y), (z*x, -x*z), (z*y, -y*z)]
>>> sorted(E.cover().kernel().gens(),key=str)
[x^2, y^2, z^2]
>>> E.inject_variables()
Defining xbar, ybar, zbar
>>> x,y,z = (xbar,ybar,zbar)
>>> y*x
-x*y
>>> all(v**Integer(2)==Integer(0) for v in E.gens())
True
>>> E.one()
1
class sage.rings.polynomial.plural.ExteriorAlgebra_plural[source]#

Bases: NCPolynomialRing_plural

class sage.rings.polynomial.plural.G_AlgFactory[source]#

Bases: UniqueFactory

A factory for the creation of g-algebras as unique parents.

create_key_and_extra_args(base_ring, c, d, names=None, order=None, category=None, check=None)[source]#

Create a unique key for g-algebras.

INPUT:

  • base_ring – a ring

  • c, d – two matrices

  • names – a tuple or list of names

  • order – (optional) term order

  • category – (optional) category

  • check – optional bool

create_object(version, key, **extra_args)[source]#

Create a g-algebra to a given unique key.

INPUT:

  • key – a 6-tuple, formed by a base ring, a tuple of names, two matrices over a polynomial ring over the base ring with the given variable names, a term order, and a category

  • extra_args – a dictionary, whose only relevant key is ‘check’.

class sage.rings.polynomial.plural.NCPolynomialRing_plural[source]#

Bases: Ring

A non-commutative polynomial ring.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y})
sage: H._is_category_initialized()
True
sage: H.category()
Category of algebras over Rational Field
sage: TestSuite(H).run()

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> H = A.g_algebra({y*x:x*y-z, z*x:x*z+Integer(2)*x, z*y:y*z-Integer(2)*y})
>>> H._is_category_initialized()
True
>>> H.category()
Category of algebras over Rational Field
>>> TestSuite(H).run()

Note that two variables commute if they are not part of the given relations:

sage: H.<x,y,z> = A.g_algebra({z*x:x*z+2*x, z*y:y*z-2*y})
sage: x*y == y*x
True

>>> from sage.all import *
>>> H = A.g_algebra({z*x:x*z+Integer(2)*x, z*y:y*z-Integer(2)*y}, names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> x*y == y*x
True
algebra_generators()[source]#

Return the algebra generators of self.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P = A.g_algebra(relations={y*x:-x*y}, order='lex')
sage: P.algebra_generators()
Finite family {'x': x, 'y': y, 'z': z}

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y}, order='lex')
>>> P.algebra_generators()
Finite family {'x': x, 'y': y, 'z': z}
free_algebra()[source]#

Return the free algebra of which this is the quotient.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex')
sage: B = P.free_algebra()
sage: A == B
True

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y}, order = 'lex')
>>> B = P.free_algebra()
>>> A == B
True
gen(n=0)[source]#

Return the n-th generator of this noncommutative polynomial ring.

INPUT:

  • n – an integer >= 0

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex')
sage: P.gen(),P.gen(1)
(x, y)

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y}, order = 'lex')
>>> P.gen(),P.gen(Integer(1))
(x, y)

Note that the generators are not cached:

sage: P.gen(1) is P.gen(1)
False

>>> from sage.all import *
>>> P.gen(Integer(1)) is P.gen(Integer(1))
False
ideal(*gens, **kwds)[source]#

Create an ideal in this polynomial ring.

INPUT:

  • *gens – list or tuple of generators (or several input arguments)

  • coerce – bool (default: True); this must be a keyword argument. Only set it to False if you are certain that each generator is already in the ring.

  • side – string (either “left”, which is the default, or “twosided”) Must be a keyword argument. Defines whether the ideal is a left ideal or a two-sided ideal. Right ideals are not implemented.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P.<x,y,z> = A.g_algebra(relations={y*x:-x*y}, order = 'lex')

sage: P.ideal([x + 2*y + 2*z-1, 2*x*y + 2*y*z-y, x^2 + 2*y^2 + 2*z^2-x])
Left Ideal (x + 2*y + 2*z - 1, 2*x*y + 2*y*z - y, x^2 - x + 2*y^2 + 2*z^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -x*y}
sage: P.ideal([x + 2*y + 2*z-1, 2*x*y + 2*y*z-y, x^2 + 2*y^2 + 2*z^2-x], side="twosided")
Twosided Ideal (x + 2*y + 2*z - 1, 2*x*y + 2*y*z - y, x^2 - x + 2*y^2 + 2*z^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -x*y}

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y}, order = 'lex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)

>>> P.ideal([x + Integer(2)*y + Integer(2)*z-Integer(1), Integer(2)*x*y + Integer(2)*y*z-y, x**Integer(2) + Integer(2)*y**Integer(2) + Integer(2)*z**Integer(2)-x])
Left Ideal (x + 2*y + 2*z - 1, 2*x*y + 2*y*z - y, x^2 - x + 2*y^2 + 2*z^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -x*y}
>>> P.ideal([x + Integer(2)*y + Integer(2)*z-Integer(1), Integer(2)*x*y + Integer(2)*y*z-y, x**Integer(2) + Integer(2)*y**Integer(2) + Integer(2)*z**Integer(2)-x], side="twosided")
Twosided Ideal (x + 2*y + 2*z - 1, 2*x*y + 2*y*z - y, x^2 - x + 2*y^2 + 2*z^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -x*y}
is_commutative()[source]#

Return False.

Todo

Provide a mathematically correct answer.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex')
sage: P.is_commutative()
False

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y}, order = 'lex')
>>> P.is_commutative()
False
is_field(*args, **kwargs)[source]#

Return False.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex')
sage: P.is_field()
False

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y}, order = 'lex')
>>> P.is_field()
False
monomial_all_divisors(t)[source]#

Return a list of all monomials that divide t.

Coefficients are ignored.

INPUT:

  • t – a monomial

OUTPUT:

a list of monomials

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P = A.g_algebra(relations={y*x:-x*y},  order='lex')
sage: P.inject_variables()
Defining x, y, z

sage: P.monomial_all_divisors(x^2*z^3)
[x, x^2, z, x*z, x^2*z, z^2, x*z^2, x^2*z^2, z^3, x*z^3, x^2*z^3]

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y},  order='lex')
>>> P.inject_variables()
Defining x, y, z

>>> P.monomial_all_divisors(x**Integer(2)*z**Integer(3))
[x, x^2, z, x*z, x^2*z, z^2, x*z^2, x^2*z^2, z^3, x*z^3, x^2*z^3]

ALGORITHM: addwithcarry idea by Toon Segers

monomial_divides(a, b)[source]#

Return False if a does not divide b and True otherwise.

Coefficients are ignored.

INPUT:

  • a – monomial

  • b – monomial

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P = A.g_algebra(relations={y*x:-x*y},  order='lex')
sage: P.inject_variables()
Defining x, y, z

sage: P.monomial_divides(x*y*z, x^3*y^2*z^4)
True
sage: P.monomial_divides(x^3*y^2*z^4, x*y*z)
False

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y},  order='lex')
>>> P.inject_variables()
Defining x, y, z

>>> P.monomial_divides(x*y*z, x**Integer(3)*y**Integer(2)*z**Integer(4))
True
>>> P.monomial_divides(x**Integer(3)*y**Integer(2)*z**Integer(4), x*y*z)
False
monomial_lcm(f, g)[source]#

LCM for monomials. Coefficients are ignored.

INPUT:

  • f – monomial

  • g – monomial

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P = A.g_algebra(relations={y*x:-x*y},  order='lex')
sage: P.inject_variables()
Defining x, y, z

sage: P.monomial_lcm(3/2*x*y,x)
x*y

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y},  order='lex')
>>> P.inject_variables()
Defining x, y, z

>>> P.monomial_lcm(Integer(3)/Integer(2)*x*y,x)
x*y
monomial_pairwise_prime(g, h)[source]#

Return True if h and g are pairwise prime.

Both h and g are treated as monomials.

Coefficients are ignored.

INPUT:

  • h – monomial

  • g – monomial

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P = A.g_algebra(relations={y*x:-x*y},  order='lex')
sage: P.inject_variables()
Defining x, y, z

sage: P.monomial_pairwise_prime(x^2*z^3, y^4)
True

sage: P.monomial_pairwise_prime(1/2*x^3*y^2, 3/4*y^3)
False

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y},  order='lex')
>>> P.inject_variables()
Defining x, y, z

>>> P.monomial_pairwise_prime(x**Integer(2)*z**Integer(3), y**Integer(4))
True

>>> P.monomial_pairwise_prime(Integer(1)/Integer(2)*x**Integer(3)*y**Integer(2), Integer(3)/Integer(4)*y**Integer(3))
False
monomial_quotient(f, g, coeff=False)[source]#

Return f/g, where both f and g are treated as monomials.

Coefficients are ignored by default.

INPUT:

  • f – monomial

  • g – monomial

  • coeff – divide coefficients as well (default: False)

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P = A.g_algebra(relations={y*x:-x*y},  order='lex')
sage: P.inject_variables()
Defining x, y, z

sage: P.monomial_quotient(3/2*x*y,x,coeff=True)
3/2*y

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y},  order='lex')
>>> P.inject_variables()
Defining x, y, z

>>> P.monomial_quotient(Integer(3)/Integer(2)*x*y,x,coeff=True)
3/2*y

Note that \(\ZZ\) behaves differently if coeff=True:

sage: P.monomial_quotient(2*x,3*x)
1
sage: P.monomial_quotient(2*x,3*x,coeff=True)
2/3

>>> from sage.all import *
>>> P.monomial_quotient(Integer(2)*x,Integer(3)*x)
1
>>> P.monomial_quotient(Integer(2)*x,Integer(3)*x,coeff=True)
2/3

Warning

Assumes that the head term of f is a multiple of the head term of g and return the multiplicant m. If this rule is violated, funny things may happen.

monomial_reduce(f, G)[source]#

Try to find a g in G where g.lm() divides f. If found (flt,g) is returned, (0,0) otherwise, where flt is f/g.lm().

It is assumed that G is iterable and contains only elements in this polynomial ring.

Coefficients are ignored.

INPUT:

  • f – monomial

  • G – list/set of mpolynomials

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P = A.g_algebra(relations={y*x:-x*y},  order='lex')
sage: P.inject_variables()
Defining x, y, z

sage: f = x*y^2
sage: G = [ 3/2*x^3 + y^2 + 1/2, 1/4*x*y + 2/7, 1/2  ]
sage: P.monomial_reduce(f,G)
(y, 1/4*x*y + 2/7)

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y},  order='lex')
>>> P.inject_variables()
Defining x, y, z

>>> f = x*y**Integer(2)
>>> G = [ Integer(3)/Integer(2)*x**Integer(3) + y**Integer(2) + Integer(1)/Integer(2), Integer(1)/Integer(4)*x*y + Integer(2)/Integer(7), Integer(1)/Integer(2)  ]
>>> P.monomial_reduce(f,G)
(y, 1/4*x*y + 2/7)
ngens()[source]#

Return the number of variables in this noncommutative polynomial ring.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P.<x,y,z> = A.g_algebra(relations={y*x:-x*y}, order = 'lex')
sage: P.ngens()
3

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y}, order = 'lex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> P.ngens()
3
relations(add_commutative=False)[source]#

Return the relations of this g-algebra.

INPUT:

add_commutative (optional bool, default False)

OUTPUT:

The defining relations. There are some implicit relations: Two generators commute if they are not part of any given relation. The implicit relations are not provided, unless add_commutative==True.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H.<x,y,z> = A.g_algebra({z*x:x*z+2*x, z*y:y*z-2*y})
sage: x*y == y*x
True
sage: H.relations()
{z*x: x*z + 2*x, z*y: y*z - 2*y}
sage: H.relations(add_commutative=True)
{y*x: x*y, z*x: x*z + 2*x, z*y: y*z - 2*y}

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> H = A.g_algebra({z*x:x*z+Integer(2)*x, z*y:y*z-Integer(2)*y}, names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> x*y == y*x
True
>>> H.relations()
{z*x: x*z + 2*x, z*y: y*z - 2*y}
>>> H.relations(add_commutative=True)
{y*x: x*y, z*x: x*z + 2*x, z*y: y*z - 2*y}
term_order()[source]#

Return the term ordering of the noncommutative ring.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex')
sage: P.term_order()
Lexicographic term order

sage: P = A.g_algebra(relations={y*x:-x*y})
sage: P.term_order()
Degree reverse lexicographic term order

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y}, order = 'lex')
>>> P.term_order()
Lexicographic term order

>>> P = A.g_algebra(relations={y*x:-x*y})
>>> P.term_order()
Degree reverse lexicographic term order
class sage.rings.polynomial.plural.NCPolynomial_plural[source]#

Bases: RingElement

A noncommutative multivariate polynomial implemented using libSINGULAR.

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 this element.

Note

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

EXAMPLES:

sage: A.<x,z,y> = FreeAlgebra(QQ, 3)
sage: R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: R.inject_variables()
Defining x, z, y
sage: f=x*y+y+5
sage: f.coefficient({x:0,y:1})
1
sage: f.coefficient({x:0})
y + 5
sage: f=(1+y+y^2)*(1+x+x^2)
sage: f.coefficient({x:0})
z + y^2 + y + 1

sage: f.coefficient(x)
y^2 - y + 1

sage: f.coefficient([0,None]) # not tested
y^2 + y + 1

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'z', 'y',)); (x, z, y,) = A._first_ngens(3)
>>> R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> R.inject_variables()
Defining x, z, y
>>> f=x*y+y+Integer(5)
>>> f.coefficient({x:Integer(0),y:Integer(1)})
1
>>> f.coefficient({x:Integer(0)})
y + 5
>>> f=(Integer(1)+y+y**Integer(2))*(Integer(1)+x+x**Integer(2))
>>> f.coefficient({x:Integer(0)})
z + y^2 + y + 1

>>> f.coefficient(x)
y^2 - y + 1

>>> f.coefficient([Integer(0),None]) # not tested
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.

sage: f.coefficient(x^0) # outputs the full polynomial
x^2*y^2 + x^2*y + x^2 + x*y^2 - x*y + x + z + y^2 + y + 1

sage: A.<x,z,y> = FreeAlgebra(GF(389), 3)
sage: R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: R.inject_variables()
Defining x, z, y
sage: f=x*y+5
sage: c=f.coefficient({x:0,y:0}); c
5
sage: parent(c)
Noncommutative Multivariate Polynomial Ring in x, z, y over Finite Field of size 389, nc-relations: {y*x: -x*y + z}

>>> from sage.all import *
>>> f.coefficient(x**Integer(0)) # outputs the full polynomial
x^2*y^2 + x^2*y + x^2 + x*y^2 - x*y + x + z + y^2 + y + 1

>>> A = FreeAlgebra(GF(Integer(389)), Integer(3), names=('x', 'z', 'y',)); (x, z, y,) = A._first_ngens(3)
>>> R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> R.inject_variables()
Defining x, z, y
>>> f=x*y+Integer(5)
>>> c=f.coefficient({x:Integer(0),y:Integer(0)}); c
5
>>> parent(c)
Noncommutative Multivariate Polynomial Ring in x, z, y over Finite Field of size 389, nc-relations: {y*x: -x*y + z}

AUTHOR:

  • Joel B. Mohler (2007-10-31)

constant_coefficient()[source]#

Return the constant coefficient of this multivariate polynomial.

EXAMPLES:

sage: A.<x,z,y> = FreeAlgebra(GF(389), 3)
sage: P = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: P.inject_variables()
Defining x, z, y
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 *
>>> A = FreeAlgebra(GF(Integer(389)), Integer(3), names=('x', 'z', 'y',)); (x, z, y,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> P.inject_variables()
Defining x, z, y
>>> 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)[source]#

Return the maximal degree of this polynomial in x, where x must be one of the generators for the parent of this polynomial.

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.

OUTPUT:

integer

EXAMPLES:

sage: A.<x,z,y> = FreeAlgebra(QQ, 3)
sage: R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: R.inject_variables()
Defining x, z, y
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 *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'z', 'y',)); (x, z, y,) = A._first_ngens(3)
>>> R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> R.inject_variables()
Defining x, z, y
>>> 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
degrees()[source]#

Return a tuple with the maximal degree of each variable in this polynomial.

The list of degrees is ordered by the order of the generators.

EXAMPLES:

sage: A.<y0,y1,y2> = FreeAlgebra(QQ, 3)
sage: R = A.g_algebra(relations={y1*y0:-y0*y1 + y2},  order='lex')
sage: R.inject_variables()
Defining y0, y1, y2
sage: q = 3*y0*y1*y1*y2; q
3*y0*y1^2*y2
sage: q.degrees()
(1, 2, 1)
sage: (q + y0^5).degrees()
(5, 2, 1)

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('y0', 'y1', 'y2',)); (y0, y1, y2,) = A._first_ngens(3)
>>> R = A.g_algebra(relations={y1*y0:-y0*y1 + y2},  order='lex')
>>> R.inject_variables()
Defining y0, y1, y2
>>> q = Integer(3)*y0*y1*y1*y2; q
3*y0*y1^2*y2
>>> q.degrees()
(1, 2, 1)
>>> (q + y0**Integer(5)).degrees()
(5, 2, 1)
dict()[source]#

Return a dictionary representing self. This dictionary is in the same format as the generic MPolynomial: The dictionary consists of ETuple:coefficient pairs.

EXAMPLES:

sage: A.<x,z,y> = FreeAlgebra(GF(389), 3)
sage: R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: R.inject_variables()
Defining x, z, y

sage: f = (2*x*y^3*z^2 + (7)*x^2 + (3))
sage: f.dict()
{(0, 0, 0): 3, (1, 2, 3): 2, (2, 0, 0): 7}

>>> from sage.all import *
>>> A = FreeAlgebra(GF(Integer(389)), Integer(3), names=('x', 'z', 'y',)); (x, z, y,) = A._first_ngens(3)
>>> R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> R.inject_variables()
Defining x, z, y

>>> f = (Integer(2)*x*y**Integer(3)*z**Integer(2) + (Integer(7))*x**Integer(2) + (Integer(3)))
>>> f.dict()
{(0, 0, 0): 3, (1, 2, 3): 2, (2, 0, 0): 7}
exponents(as_ETuples=True)[source]#

Return the exponents of the monomials appearing in this polynomial.

INPUT:

  • as_ETuples – (default: True) if True returns the result as an list of ETuples otherwise returns a list of tuples

EXAMPLES:

sage: A.<x,z,y> = FreeAlgebra(GF(389), 3)
sage: R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: R.inject_variables()
Defining x, z, y
sage: f = x^3 + y + 2*z^2
sage: f.exponents()
[(3, 0, 0), (0, 2, 0), (0, 0, 1)]
sage: f.exponents(as_ETuples=False)
[(3, 0, 0), (0, 2, 0), (0, 0, 1)]

>>> from sage.all import *
>>> A = FreeAlgebra(GF(Integer(389)), Integer(3), names=('x', 'z', 'y',)); (x, z, y,) = A._first_ngens(3)
>>> R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> R.inject_variables()
Defining x, z, y
>>> f = x**Integer(3) + y + Integer(2)*z**Integer(2)
>>> f.exponents()
[(3, 0, 0), (0, 2, 0), (0, 0, 1)]
>>> f.exponents(as_ETuples=False)
[(3, 0, 0), (0, 2, 0), (0, 0, 1)]
is_constant()[source]#

Return True if this polynomial is constant.

EXAMPLES:

sage: A.<x,z,y> = FreeAlgebra(GF(389), 3)
sage: P = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: P.inject_variables()
Defining x, z, y
sage: x.is_constant()
False
sage: P(1).is_constant()
True

>>> from sage.all import *
>>> A = FreeAlgebra(GF(Integer(389)), Integer(3), names=('x', 'z', 'y',)); (x, z, y,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> P.inject_variables()
Defining x, z, y
>>> x.is_constant()
False
>>> P(Integer(1)).is_constant()
True
is_homogeneous()[source]#

Return True if this polynomial is homogeneous.

EXAMPLES:

sage: A.<x,z,y> = FreeAlgebra(GF(389), 3)
sage: P = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: P.inject_variables()
Defining x, z, y
sage: (x+y+z).is_homogeneous()
True
sage: (x.parent()(0)).is_homogeneous()
True
sage: (x+y^2+z^3).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 *
>>> A = FreeAlgebra(GF(Integer(389)), Integer(3), names=('x', 'z', 'y',)); (x, z, y,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> P.inject_variables()
Defining x, z, y
>>> (x+y+z).is_homogeneous()
True
>>> (x.parent()(Integer(0))).is_homogeneous()
True
>>> (x+y**Integer(2)+z**Integer(3)).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
is_monomial()[source]#

Return True if this polynomial is a monomial.

A monomial is defined to be a product of generators with coefficient 1.

EXAMPLES:

sage: A.<x,z,y> = FreeAlgebra(GF(389), 3)
sage: P = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: P.inject_variables()
Defining x, z, y
sage: x.is_monomial()
True
sage: (2*x).is_monomial()
False
sage: (x*y).is_monomial()
True
sage: (x*y + x).is_monomial()
False

>>> from sage.all import *
>>> A = FreeAlgebra(GF(Integer(389)), Integer(3), names=('x', 'z', 'y',)); (x, z, y,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> P.inject_variables()
Defining x, z, y
>>> x.is_monomial()
True
>>> (Integer(2)*x).is_monomial()
False
>>> (x*y).is_monomial()
True
>>> (x*y + x).is_monomial()
False
is_zero()[source]#

Return True if this polynomial is zero.

EXAMPLES:

sage: A.<x,z,y> = FreeAlgebra(QQ, 3)
sage: R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: R.inject_variables()
Defining x, z, y

sage: x.is_zero()
False
sage: (x-x).is_zero()
True

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'z', 'y',)); (x, z, y,) = A._first_ngens(3)
>>> R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> R.inject_variables()
Defining x, z, y

>>> x.is_zero()
False
>>> (x-x).is_zero()
True
lc()[source]#

Leading coefficient of this polynomial with respect to the term order of self.parent().

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(GF(7), 3)
sage: R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: R.inject_variables()
Defining x, y, z

sage: f = 3*x^1*y^2 + 2*y^3*z^4
sage: f.lc()
3

sage: f = 5*x^3*y^2*z^4 + 4*x^3*y^2*z^1
sage: f.lc()
5

>>> from sage.all import *
>>> A = FreeAlgebra(GF(Integer(7)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> R.inject_variables()
Defining x, y, z

>>> f = Integer(3)*x**Integer(1)*y**Integer(2) + Integer(2)*y**Integer(3)*z**Integer(4)
>>> f.lc()
3

>>> f = Integer(5)*x**Integer(3)*y**Integer(2)*z**Integer(4) + Integer(4)*x**Integer(3)*y**Integer(2)*z**Integer(1)
>>> f.lc()
5
lm()[source]#

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

In Sage, a monomial is a product of variables in some power without a coefficient.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(GF(7), 3)
sage: R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: R.inject_variables()
Defining x, y, z
sage: f = x^1*y^2 + y^3*z^4
sage: f.lm()
x*y^2
sage: f = x^3*y^2*z^4 + x^3*y^2*z^1
sage: f.lm()
x^3*y^2*z^4

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: R = A.g_algebra(relations={y*x:-x*y + z},  order='deglex')
sage: R.inject_variables()
Defining x, y, z
sage: f = x^1*y^2*z^3 + x^3*y^2*z^0
sage: f.lm()
x*y^2*z^3
sage: f = x^1*y^2*z^4 + x^1*y^1*z^5
sage: f.lm()
x*y^2*z^4

sage: A.<x,y,z> = FreeAlgebra(GF(127), 3)
sage: R = A.g_algebra(relations={y*x:-x*y + z},  order='degrevlex')
sage: R.inject_variables()
Defining x, y, z
sage: f = x^1*y^5*z^2 + x^4*y^1*z^3
sage: f.lm()
x*y^5*z^2
sage: f = x^4*y^7*z^1 + x^4*y^2*z^3
sage: f.lm()
x^4*y^7*z

>>> from sage.all import *
>>> A = FreeAlgebra(GF(Integer(7)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> R.inject_variables()
Defining x, y, z
>>> f = x**Integer(1)*y**Integer(2) + y**Integer(3)*z**Integer(4)
>>> f.lm()
x*y^2
>>> f = x**Integer(3)*y**Integer(2)*z**Integer(4) + x**Integer(3)*y**Integer(2)*z**Integer(1)
>>> f.lm()
x^3*y^2*z^4

>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> R = A.g_algebra(relations={y*x:-x*y + z},  order='deglex')
>>> R.inject_variables()
Defining x, y, z
>>> f = x**Integer(1)*y**Integer(2)*z**Integer(3) + x**Integer(3)*y**Integer(2)*z**Integer(0)
>>> f.lm()
x*y^2*z^3
>>> f = x**Integer(1)*y**Integer(2)*z**Integer(4) + x**Integer(1)*y**Integer(1)*z**Integer(5)
>>> f.lm()
x*y^2*z^4

>>> A = FreeAlgebra(GF(Integer(127)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> R = A.g_algebra(relations={y*x:-x*y + z},  order='degrevlex')
>>> R.inject_variables()
Defining x, y, z
>>> f = x**Integer(1)*y**Integer(5)*z**Integer(2) + x**Integer(4)*y**Integer(1)*z**Integer(3)
>>> f.lm()
x*y^5*z^2
>>> f = x**Integer(4)*y**Integer(7)*z**Integer(1) + x**Integer(4)*y**Integer(2)*z**Integer(3)
>>> f.lm()
x^4*y^7*z
lt()[source]#

Return the leading term of this polynomial.

In Sage, a term is a product of variables in some power and a coefficient.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(GF(7), 3)
sage: R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: R.inject_variables()
Defining x, y, z

sage: f = 3*x^1*y^2 + 2*y^3*z^4
sage: f.lt()
3*x*y^2

sage: f = 5*x^3*y^2*z^4 + 4*x^3*y^2*z^1
sage: f.lt()
-2*x^3*y^2*z^4

>>> from sage.all import *
>>> A = FreeAlgebra(GF(Integer(7)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> R.inject_variables()
Defining x, y, z

>>> f = Integer(3)*x**Integer(1)*y**Integer(2) + Integer(2)*y**Integer(3)*z**Integer(4)
>>> f.lt()
3*x*y^2

>>> f = Integer(5)*x**Integer(3)*y**Integer(2)*z**Integer(4) + Integer(4)*x**Integer(3)*y**Integer(2)*z**Integer(1)
>>> f.lt()
-2*x^3*y^2*z^4
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: A.<x,z,y> = FreeAlgebra(GF(389), 3)
sage: P = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: P.inject_variables()
Defining x, z, y

The parent of the return is a member of the base ring.
sage: f = 2 * x * y
sage: c = f.monomial_coefficient(x*y); c
2
sage: c.parent()
Finite Field of size 389

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)
388
sage: f.monomial_coefficient(x^10)
0

>>> from sage.all import *
>>> A = FreeAlgebra(GF(Integer(389)), Integer(3), names=('x', 'z', 'y',)); (x, z, y,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> P.inject_variables()
Defining x, z, y

The parent of the return is a member of the base ring.
>>> f = Integer(2) * x * y
>>> c = f.monomial_coefficient(x*y); c
2
>>> c.parent()
Finite Field of size 389

>>> 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))
388
>>> f.monomial_coefficient(x**Integer(10))
0
monomial_coefficients(copy=True)[source]#

Return a dictionary representation of self with the keys the exponent vectors and the values the corresponding coefficients.

INPUT:

  • “copy” – ignored

EXAMPLES:

sage: A.<x,z,y> = FreeAlgebra(GF(389), 3)
sage: R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: R.inject_variables()
Defining x, z, y
sage: f = (2*x*y^3*z^2 + (7)*x^2 + (3))
sage: d = f.monomial_coefficients(False); d
{(0, 0, 0): 3, (1, 2, 3): 2, (2, 0, 0): 7}
sage: d.clear()
sage: f.monomial_coefficients()
{(0, 0, 0): 3, (1, 2, 3): 2, (2, 0, 0): 7}

>>> from sage.all import *
>>> A = FreeAlgebra(GF(Integer(389)), Integer(3), names=('x', 'z', 'y',)); (x, z, y,) = A._first_ngens(3)
>>> R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> R.inject_variables()
Defining x, z, y
>>> f = (Integer(2)*x*y**Integer(3)*z**Integer(2) + (Integer(7))*x**Integer(2) + (Integer(3)))
>>> d = f.monomial_coefficients(False); d
{(0, 0, 0): 3, (1, 2, 3): 2, (2, 0, 0): 7}
>>> d.clear()
>>> f.monomial_coefficients()
{(0, 0, 0): 3, (1, 2, 3): 2, (2, 0, 0): 7}
monomials()[source]#

Return the list of monomials in self

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

EXAMPLES:

sage: A.<x,z,y> = FreeAlgebra(GF(389), 3)
sage: P = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: P.inject_variables()
Defining x, z, y
sage: f = x + (3*2)*y*z^2 + (2+3)
sage: f.monomials()
[x, z^2*y, 1]
sage: f = P(3^2)
sage: f.monomials()
[1]

>>> from sage.all import *
>>> A = FreeAlgebra(GF(Integer(389)), Integer(3), names=('x', 'z', 'y',)); (x, z, y,) = A._first_ngens(3)
>>> P = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> P.inject_variables()
Defining x, z, y
>>> f = x + (Integer(3)*Integer(2))*y*z**Integer(2) + (Integer(2)+Integer(3))
>>> f.monomials()
[x, z^2*y, 1]
>>> f = P(Integer(3)**Integer(2))
>>> f.monomials()
[1]
reduce(I)[source]#

EXAMPLES:

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

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> H = A.g_algebra({y*x:x*y-z, z*x:x*z+Integer(2)*x, z*y:y*z-Integer(2)*y}, names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> I = H.ideal([y**Integer(2), x**Integer(2), z**Integer(2)-H.one()],coerce=False)

The result of reduction is not the normal form, if one reduces by a list of polynomials:

sage: (x*z).reduce(I.gens())
x*z

>>> from sage.all import *
>>> (x*z).reduce(I.gens())
x*z

However, if the argument is an ideal, then a normal form (reduction with respect to a two-sided Groebner basis) is returned:

sage: (x*z).reduce(I)
-x

>>> from sage.all import *
>>> (x*z).reduce(I)
-x

The Groebner basis shows that the result is correct:

sage: I.std() #random
Left Ideal (z^2 - 1, y*z - y, x*z + x, y^2, 2*x*y - z - 1, x^2) of
Noncommutative Multivariate Polynomial Ring in x, y, z over Rational
Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
sage: sorted(I.std().gens(),key=str)
[2*x*y - z - 1, x*z + x, x^2, y*z - y, y^2, z^2 - 1]

>>> from sage.all import *
>>> I.std() #random
Left Ideal (z^2 - 1, y*z - y, x*z + x, y^2, 2*x*y - z - 1, x^2) of
Noncommutative Multivariate Polynomial Ring in x, y, z over Rational
Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
>>> sorted(I.std().gens(),key=str)
[2*x*y - z - 1, x*z + x, x^2, y*z - y, y^2, z^2 - 1]
total_degree()[source]#

Return the total degree of self, which is the maximum degree of all monomials in self.

EXAMPLES:

sage: A.<x,z,y> = FreeAlgebra(QQ, 3)
sage: R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
sage: R.inject_variables()
Defining x, z, y
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 *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'z', 'y',)); (x, z, y,) = A._first_ngens(3)
>>> R = A.g_algebra(relations={y*x:-x*y + z},  order='lex')
>>> R.inject_variables()
Defining x, z, y
>>> 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
sage.rings.polynomial.plural.SCA(base_ring, names, alt_vars, order='degrevlex')[source]#

Return a free graded-commutative algebra

This is also known as a free super-commutative algebra.

INPUT:

  • base_ring – the ground field

  • names – a list of variable names

  • alt_vars – a list of indices of to be anti-commutative variables (odd variables)

  • order – ordering to be used for the constructed algebra

EXAMPLES:

sage: from sage.rings.polynomial.plural import SCA
sage: E = SCA(QQ, ['x', 'y', 'z'], [0, 1], order = 'degrevlex')
sage: E
Quotient of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -x*y} by the ideal (y^2, x^2)
sage: E.inject_variables()
Defining xbar, ybar, zbar
sage: x,y,z = (xbar,ybar,zbar)
sage: y*x
-x*y
sage: z*x
x*z
sage: x^2
0
sage: y^2
0
sage: z^2
z^2
sage: E.one()
1

>>> from sage.all import *
>>> from sage.rings.polynomial.plural import SCA
>>> E = SCA(QQ, ['x', 'y', 'z'], [Integer(0), Integer(1)], order = 'degrevlex')
>>> E
Quotient of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -x*y} by the ideal (y^2, x^2)
>>> E.inject_variables()
Defining xbar, ybar, zbar
>>> x,y,z = (xbar,ybar,zbar)
>>> y*x
-x*y
>>> z*x
x*z
>>> x**Integer(2)
0
>>> y**Integer(2)
0
>>> z**Integer(2)
z^2
>>> E.one()
1
sage.rings.polynomial.plural.new_CRing(rw, base_ring)[source]#

Construct MPolynomialRing_libsingular from ringWrap, assuming the ground field to be base_ring

EXAMPLES:

sage: H.<x,y,z> = PolynomialRing(QQ, 3)
sage: from sage.libs.singular.function import singular_function

sage: ringlist = singular_function('ringlist')
sage: ring = singular_function("ring")

sage: L = ringlist(H, ring=H); L
[0, ['x', 'y', 'z'], [['dp', (1, 1, 1)], ['C', (0,)]], [0]]

sage: len(L)
4

sage: W = ring(L, ring=H); W
<RingWrap>

sage: from sage.rings.polynomial.plural import new_CRing
sage: R = new_CRing(W, H.base_ring())
sage: R # indirect doctest
Multivariate Polynomial Ring in x, y, z over Rational Field

>>> from sage.all import *
>>> H = PolynomialRing(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> from sage.libs.singular.function import singular_function

>>> ringlist = singular_function('ringlist')
>>> ring = singular_function("ring")

>>> L = ringlist(H, ring=H); L
[0, ['x', 'y', 'z'], [['dp', (1, 1, 1)], ['C', (0,)]], [0]]

>>> len(L)
4

>>> W = ring(L, ring=H); W
<RingWrap>

>>> from sage.rings.polynomial.plural import new_CRing
>>> R = new_CRing(W, H.base_ring())
>>> R # indirect doctest
Multivariate Polynomial Ring in x, y, z over Rational Field

Check that Issue #13145 has been resolved:

sage: h = hash(R.gen() + 1) # sets currRing
sage: from sage.libs.singular.ring import ring_refcount_dict, currRing_wrapper
sage: curcnt = ring_refcount_dict[currRing_wrapper()]
sage: newR = new_CRing(W, H.base_ring())
sage: ring_refcount_dict[currRing_wrapper()] - curcnt
2

>>> from sage.all import *
>>> h = hash(R.gen() + Integer(1)) # sets currRing
>>> from sage.libs.singular.ring import ring_refcount_dict, currRing_wrapper
>>> curcnt = ring_refcount_dict[currRing_wrapper()]
>>> newR = new_CRing(W, H.base_ring())
>>> ring_refcount_dict[currRing_wrapper()] - curcnt
2

Check that Issue #29311 is fixed:

sage: R.<x,y,z> = QQ[]
sage: from sage.libs.singular.function_factory import ff
sage: W = ff.ring(ff.ringlist(R), ring=R)
sage: C = sage.rings.polynomial.plural.new_CRing(W, R.base_ring())
sage: C.one()
1

>>> from sage.all import *
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> from sage.libs.singular.function_factory import ff
>>> W = ff.ring(ff.ringlist(R), ring=R)
>>> C = sage.rings.polynomial.plural.new_CRing(W, R.base_ring())
>>> C.one()
1
sage.rings.polynomial.plural.new_NRing(rw, base_ring)[source]#

Construct NCPolynomialRing_plural from ringWrap, assuming the ground field to be base_ring

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H = A.g_algebra({y*x:x*y-1})
sage: H.inject_variables()
Defining x, y, z
sage: z*x
x*z
sage: z*y
y*z
sage: y*x
x*y - 1
sage: I = H.ideal([y^2, x^2, z^2-1])
sage: I._groebner_basis_libsingular()
[1]

sage: from sage.libs.singular.function import singular_function

sage: ringlist = singular_function('ringlist')
sage: ring = singular_function("ring")

sage: L = ringlist(H, ring=H); L
[
                                                           [0 1 1]
                                                           [0 0 1]
0, ['x', 'y', 'z'], [['dp', (1, 1, 1)], ['C', (0,)]], [0], [0 0 0],

[ 0 -1  0]
[ 0  0  0]
[ 0  0  0]
]
sage: len(L)
6

sage: W = ring(L, ring=H); W
<noncommutative RingWrap>

sage: from sage.rings.polynomial.plural import new_NRing
sage: R = new_NRing(W, H.base_ring())
sage: R # indirect doctest
Noncommutative Multivariate Polynomial Ring in x, y, z over
Rational Field, nc-relations: {y*x: x*y - 1}

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> H = A.g_algebra({y*x:x*y-Integer(1)})
>>> H.inject_variables()
Defining x, y, z
>>> z*x
x*z
>>> z*y
y*z
>>> y*x
x*y - 1
>>> I = H.ideal([y**Integer(2), x**Integer(2), z**Integer(2)-Integer(1)])
>>> I._groebner_basis_libsingular()
[1]

>>> from sage.libs.singular.function import singular_function

>>> ringlist = singular_function('ringlist')
>>> ring = singular_function("ring")

>>> L = ringlist(H, ring=H); L
[
                                                           [0 1 1]
                                                           [0 0 1]
0, ['x', 'y', 'z'], [['dp', (1, 1, 1)], ['C', (0,)]], [0], [0 0 0],
<BLANKLINE>
[ 0 -1  0]
[ 0  0  0]
[ 0  0  0]
]
>>> len(L)
6

>>> W = ring(L, ring=H); W
<noncommutative RingWrap>

>>> from sage.rings.polynomial.plural import new_NRing
>>> R = new_NRing(W, H.base_ring())
>>> R # indirect doctest
Noncommutative Multivariate Polynomial Ring in x, y, z over
Rational Field, nc-relations: {y*x: x*y - 1}
sage.rings.polynomial.plural.new_Ring(rw, base_ring)[source]#

Constructs a Sage ring out of low level RingWrap, which wraps a pointer to a Singular ring.

The constructed ring is either commutative or noncommutative depending on the Singular ring.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H = A.g_algebra({y*x:x*y-1})
sage: H.inject_variables()
Defining x, y, z
sage: z*x
x*z
sage: z*y
y*z
sage: y*x
x*y - 1
sage: I = H.ideal([y^2, x^2, z^2-1])
sage: I._groebner_basis_libsingular()
[1]

sage: from sage.libs.singular.function import singular_function

sage: ringlist = singular_function('ringlist')
sage: ring = singular_function("ring")

sage: L = ringlist(H, ring=H); L
[
                                                           [0 1 1]
                                                           [0 0 1]
0, ['x', 'y', 'z'], [['dp', (1, 1, 1)], ['C', (0,)]], [0], [0 0 0],

[ 0 -1  0]
[ 0  0  0]
[ 0  0  0]
]
sage: len(L)
6

sage: W = ring(L, ring=H); W
<noncommutative RingWrap>

sage: from sage.rings.polynomial.plural import new_Ring
sage: R = new_Ring(W, H.base_ring()); R
Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: x*y - 1}

>>> from sage.all import *
>>> A = FreeAlgebra(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
>>> H = A.g_algebra({y*x:x*y-Integer(1)})
>>> H.inject_variables()
Defining x, y, z
>>> z*x
x*z
>>> z*y
y*z
>>> y*x
x*y - 1
>>> I = H.ideal([y**Integer(2), x**Integer(2), z**Integer(2)-Integer(1)])
>>> I._groebner_basis_libsingular()
[1]

>>> from sage.libs.singular.function import singular_function

>>> ringlist = singular_function('ringlist')
>>> ring = singular_function("ring")

>>> L = ringlist(H, ring=H); L
[
                                                           [0 1 1]
                                                           [0 0 1]
0, ['x', 'y', 'z'], [['dp', (1, 1, 1)], ['C', (0,)]], [0], [0 0 0],
<BLANKLINE>
[ 0 -1  0]
[ 0  0  0]
[ 0  0  0]
]
>>> len(L)
6

>>> W = ring(L, ring=H); W
<noncommutative RingWrap>

>>> from sage.rings.polynomial.plural import new_Ring
>>> R = new_Ring(W, H.base_ring()); R
Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: x*y - 1}
sage.rings.polynomial.plural.unpickle_NCPolynomial_plural(R, d)[source]#

Auxiliary function to unpickle a non-commutative polynomial.