Multivariate Polynomial Rings over Generic Rings¶
Sage implements multivariate polynomial rings through several
backends. This generic implementation uses the classes PolyDict
and ETuple
to construct a dictionary with exponent tuples as keys
and coefficients as values.
AUTHORS:
- David Joyner and William Stein
- Kiran S. Kedlaya (2006-02-12): added Macaulay2 analogues of Singular features
- Martin Albrecht (2006-04-21): reorganize class hierarchy for singular rep
- Martin Albrecht (2007-04-20): reorganized class hierarchy to support Pyrex implementations
- Robert Bradshaw (2007-08-15): Coercions from rings in a subset of the variables.
EXAMPLES:
We construct the Frobenius morphism on over
:
sage: R.<x,y,z> = GF(5)[]
sage: frob = R.hom([x^5, y^5, z^5])
sage: frob(x^2 + 2*y - z^4)
-z^20 + x^10 + 2*y^5
sage: frob((x + 2*y)^3)
x^15 + x^10*y^5 + 2*x^5*y^10 - 2*y^15
sage: (x^5 + 2*y^5)^3
x^15 + x^10*y^5 + 2*x^5*y^10 - 2*y^15
We make a polynomial ring in one variable over a polynomial ring in two variables:
sage: R.<x, y> = PolynomialRing(QQ, 2)
sage: S.<t> = PowerSeriesRing(R)
sage: t*(x+y)
(x + y)*t
TESTS:
sage: PolynomialRing(GF(5), 3, 'xyz').objgens()
(Multivariate Polynomial Ring in x, y, z over Finite Field of size 5,
(x, y, z))
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class
sage.rings.polynomial.multi_polynomial_ring.
MPolynomialRing_polydict
(base_ring, n, names, order)¶ Bases:
sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_macaulay2_repr
,sage.rings.polynomial.polynomial_singular_interface.PolynomialRing_singular_repr
,sage.rings.polynomial.multi_polynomial_ring_generic.MPolynomialRing_generic
Multivariable polynomial ring.
EXAMPLES:
sage: R = PolynomialRing(Integers(12), 'x', 5); R Multivariate Polynomial Ring in x0, x1, x2, x3, x4 over Ring of integers modulo 12 sage: loads(R.dumps()) == R True
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class
sage.rings.polynomial.multi_polynomial_ring.
MPolynomialRing_polydict_domain
(base_ring, n, names, order)¶ Bases:
sage.rings.ring.IntegralDomain
,sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict
-
ideal
(*gens, **kwds)¶ Create an ideal in this polynomial ring.
-
is_field
(proof=True)¶
-
is_integral_domain
(proof=True)¶
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monomial_all_divisors
(t)¶ Return a list of all monomials that divide t, coefficients are ignored.
INPUT:
t
- a monomial
OUTPUT: a list of monomials
EXAMPLE:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex') 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]
ALGORITHM: addwithcarry idea by Toon Segers
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monomial_divides
(a, b)¶ Return False if a does not divide b and True otherwise.
INPUT:
a
- monomialb
- monomial
EXAMPLES:
sage: P.<x,y,z>=PolynomialRing(ZZ,3, order='degrevlex') 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
TESTS:
sage: P.<x,y,z>=PolynomialRing(ZZ,3, order='degrevlex') sage: P.monomial_divides(P(1), P(0)) True sage: P.monomial_divides(P(1), x) True
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monomial_lcm
(f, g)¶ LCM for monomials. Coefficients are ignored.
INPUT:
f
- monomialg
- monomial
EXAMPLE:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex') sage: P.monomial_lcm(3/2*x*y,x) x*y
TESTS:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain sage: R.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex') sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex') sage: P.monomial_lcm(x*y,R.gen()) x*y
sage: P.monomial_lcm(P(3/2),P(2/3)) 1
sage: P.monomial_lcm(x,P(1)) x
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monomial_pairwise_prime
(h, g)¶ Return True if h and g are pairwise prime. Both are treated as monomials.
INPUT:
h
- monomialg
- monomial
EXAMPLES:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex') 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
TESTS:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex') sage: Q.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex') sage: P.monomial_pairwise_prime(x^2*z^3, Q('y^4')) True
sage: P.monomial_pairwise_prime(1/2*x^3*y^2, Q(0)) True
sage: P.monomial_pairwise_prime(P(1/2),x) False
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monomial_quotient
(f, g, coeff=False)¶ Return f/g, where both f and g are treated as monomials. Coefficients are ignored by default.
INPUT:
f
- monomialg
- monomialcoeff
- divide coefficients as well (default: False)
EXAMPLE:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ, 3, order='degrevlex') sage: P.monomial_quotient(3/2*x*y,x) y
sage: P.monomial_quotient(3/2*x*y,2*x,coeff=True) 3/4*y
TESTS:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain sage: R.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex') sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex') sage: P.monomial_quotient(x*y,x) y
sage: P.monomial_quotient(x*y,R.gen()) y
sage: P.monomial_quotient(P(0),P(1)) 0
sage: P.monomial_quotient(P(1),P(0)) Traceback (most recent call last): ... ZeroDivisionError
sage: P.monomial_quotient(P(3/2),P(2/3), coeff=True) 9/4
sage: P.monomial_quotient(x,y) # Note the wrong result x*y^-1
sage: P.monomial_quotient(x,P(1)) x
Note
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.
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monomial_reduce
(f, G)¶ Try to find a g in G where g.lm() divides f. If found (g,flt) is returned, (0,0) otherwise, where flt is f/g.lm().
It is assumed that G is iterable and contains ONLY elements in self.
INPUT:
f
- monomialG
- list/set of mpolynomials
EXAMPLES:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex') sage: f = x*y^2 sage: G = [ 3/2*x^3 + y^2 + 1/2, 1/4*x*y + 2/7, P(1/2) ] sage: P.monomial_reduce(f,G) (y, 1/4*x*y + 2/7)
TESTS:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex') sage: f = x*y^2 sage: G = [ 3/2*x^3 + y^2 + 1/2, 1/4*x*y + 2/7, P(1/2) ]
sage: P.monomial_reduce(P(0),G) (0, 0)
sage: P.monomial_reduce(f,[P(0)]) (0, 0)
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