Package cc.redberry.rings.poly.multivar

Class GroebnerMethods

java.lang.Object

cc.redberry.rings.poly.multivar.GroebnerMethods

public final class GroebnerMethods
extends Object

Utility methods based on Groebner bases

Method Summary

Modifier and Type	Method	Description
static <Poly extends AMultivariatePolynomial> boolean	algebraicallyDependentQ(List<Poly> sys)	Returns true if a given set of polynomials is algebraically dependent or false otherwise.
static <Poly extends AMultivariatePolynomial> List<Poly>	algebraicRelations(List<Poly> polys)	Gives a list of algebraic relations (annihilating polynomials) for the given list of polynomials
static <Poly extends AMultivariatePolynomial> List<Poly>	eliminate(List<Poly> ideal, int variable)	Eliminates specified variables from the given ideal.
static <Poly extends AMultivariatePolynomial> List<Poly>	eliminate(List<Poly> ideal, int... variables)	Eliminates specified variables from the given ideal.
static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Poly[][]	JacobianMatrix(List<Poly> sys)	Creates a Jacobian matrix of a given list of polynomials
static <Poly extends AMultivariatePolynomial> List<Rational<Poly>>	LeinartasDecomposition(Rational<Poly> fraction)	Computes Leinartas's decomposition of given rational expression (see https://arxiv.org/abs/1206.4740)
static <Poly extends AMultivariatePolynomial> List<Poly>	NullstellensatzCertificate(List<Poly> polynomials)	Computes Nullstellensatz certificate for a given list of polynomials assuming that they have no common zeros (or equivalently assuming that the ideal formed by the list is trivial).
static <Poly extends AMultivariatePolynomial> List<Poly>	NullstellensatzCertificate(List<Poly> polynomials, boolean boundTotalDeg)	Computes Nullstellensatz certificate for a given list of polynomials assuming that they have no common zeros (or equivalently assuming that the ideal formed by the list is trivial).
static <Poly extends AMultivariatePolynomial> List<Poly>	NullstellensatzSolver(List<Poly> polynomials, Poly rhs, boolean boundTotalDeg)	Tries to find solution of the equation S_1 * f_1 + ... + S_n * f_n = g for given f_i and g and unknown S_i by transforming to a system of linear equations with unknown coefficients of S_i.
static <Poly extends AMultivariatePolynomial> boolean	probablyAlgebraicallyDependentQ(List<Poly> sys)	Returns true if a given set of polynomials is probably algebraically dependent or false otherwise (which means that the given set is certainly independent).

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Detail

eliminate

public static <Poly extends AMultivariatePolynomial> List<Poly> eliminate(List<Poly> ideal,
                                                                          int variable)

Eliminates specified variables from the given ideal.

eliminate

public static <Poly extends AMultivariatePolynomial> List<Poly> eliminate(List<Poly> ideal,
                                                                          int... variables)

Eliminates specified variables from the given ideal.

probablyAlgebraicallyDependentQ

public static <Poly extends AMultivariatePolynomial> boolean probablyAlgebraicallyDependentQ(List<Poly> sys)

Returns true if a given set of polynomials is probably algebraically dependent or false otherwise (which means that the given set is certainly independent). The method applies two criteria: it tests for lead set (LEX) independence and does a probabilistic Jacobian test.

algebraicallyDependentQ

public static <Poly extends AMultivariatePolynomial> boolean algebraicallyDependentQ(List<Poly> sys)

Returns true if a given set of polynomials is algebraically dependent or false otherwise.

algebraicRelations

public static <Poly extends AMultivariatePolynomial> List<Poly> algebraicRelations(List<Poly> polys)

Gives a list of algebraic relations (annihilating polynomials) for the given list of polynomials

JacobianMatrix

public static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Poly[][] JacobianMatrix(List<Poly> sys)

Creates a Jacobian matrix of a given list of polynomials

NullstellensatzCertificate

public static <Poly extends AMultivariatePolynomial> List<Poly> NullstellensatzCertificate(List<Poly> polynomials)

Computes Nullstellensatz certificate for a given list of polynomials assuming that they have no common zeros (or equivalently assuming that the ideal formed by the list is trivial). The method doesn't perform explicit check that the polynomials have no common zero, so if they are the method will fail.

Parameters:

polynomials - list of polynomials

Returns:

polynomials S_i such that S_1 * f_1 + ... + S_n * f_n = 1 or null if no solution with moderate degree bounds exist (either since polynomials have a common root or because the degree bound on the solutions is so big that the system is intractable for computer)

NullstellensatzCertificate

public static <Poly extends AMultivariatePolynomial> List<Poly> NullstellensatzCertificate(List<Poly> polynomials,
                                                                                           boolean boundTotalDeg)

Computes Nullstellensatz certificate for a given list of polynomials assuming that they have no common zeros (or equivalently assuming that the ideal formed by the list is trivial). The method doesn't perform explicit check that the polynomials have no common zero, so if they are the method will fail.

Parameters:

polynomials - list of polynomials

Returns:

polynomials S_i such that S_1 * f_1 + ... + S_n * f_n = 1 or null if no solution with moderate degree bounds exist (either since polynomials have a common root or because the degree bound on the solutions is so big that the system is intractable for computer)

NullstellensatzSolver

public static <Poly extends AMultivariatePolynomial> List<Poly> NullstellensatzSolver(List<Poly> polynomials,
                                                                                      Poly rhs,
                                                                                      boolean boundTotalDeg)

Tries to find solution of the equation S_1 * f_1 + ... + S_n * f_n = g for given f_i and g and unknown S_i by transforming to a system of linear equations with unknown coefficients of S_i.

Parameters:

polynomials - list of polynomials

rhs - right hand side of the equation

boundTotalDeg - whether to perform evaluations by increasing total degree of unknown polys or by increasing individual degrees of vars

Returns:

polynomials S_i such that S_1 * f_1 + ... + S_n * f_n = g or null if no solution with moderate degree bounds exists

LeinartasDecomposition

public static <Poly extends AMultivariatePolynomial> List<Rational<Poly>> LeinartasDecomposition(Rational<Poly> fraction)

Computes Leinartas's decomposition of given rational expression (see https://arxiv.org/abs/1206.4740)