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witness.h
Go to the documentation of this file.
#ifndef WITNESS_H
#define WITNESS_H
#include "polys/monomials/monomials.h"
#include "polys/simpleideals.h"
/**
* Computes a division discarding remainder of f with respect to G.
* Given f a polynomial and G={g1,...,gk} a set of polynomials in r,
* returns a matrix Q=(q1,...,qk) over r such that
* f = q1*g1+...+qk*gk+s
* is a determinate division with remainder s.
*/
matrix divisionDiscardingRemainder(const poly f, const ideal G, const ring r);
/**
* Computes a division discarding remainder of F with respect to G.
* Given F={f1,...,fl} and G={g1,...,gk} two sets of polynomials in r,
* returns a matrix Q=(qij) i=1,..,k j=1,...,l over r such that
* fj = q1j*g1+...+qkj*gk+sj
* is a determinate division with remainder sj for all j=1,...,l.
*/
matrix divisionDiscardingRemainder(const ideal F, const ideal G, const ring r);
/**
* Let w be the uppermost weight vector in the matrix defining the ordering on r.
* Let I be a Groebner basis of an ideal in r, inI its initial form with respect w.
* Given an w-homogeneous element m of inI, computes a witness g of m in I,
* i.e. g in I such that in_w(g)=m.
*/
poly witness(const poly m, const ideal I, const ideal inI, const ring r);
/**
* Computes witnesses in J for inI
* Given inI={h1,...,hl} and J={g1,...,gk} two sets of polynomials in r,
* returns a set I={f1,...,fl} of <g1,...,gk> such that
* in_w(fj)=hj for all j=1,...,l,
* where w denotes the uppoermost weight vector in the matrix defining the ordering on r.
* Assumes that hj is an element of <in_w(g1),...,in_w(gk)>
*/
ideal witness(const ideal inI, const ideal J, const ring r);
#endif
m
int m
Definition cfEzgcd.cc:128
f
FILE * f
Definition checklibs.c:9
G
STATIC_VAR TreeM * G
Definition janet.cc:31
matrix
ip_smatrix * matrix
Definition matpol.h:43
monomials.h
simpleideals.h
divisionDiscardingRemainder
matrix divisionDiscardingRemainder(const poly f, const ideal G, const ring r)
Computes a division discarding remainder of f with respect to G.
Definition witness.cc:9
witness
poly witness(const poly m, const ideal I, const ideal inI, const ring r)
Let w be the uppermost weight vector in the matrix defining the ordering on r.
Definition witness.cc:34