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witness.h
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1 #ifndef WITNESS_H
2 #define WITNESS_H
3 
5 #include "polys/simpleideals.h"
6 
7 /**
8  * Computes a division discarding remainder of f with respect to G.
9  * Given f a polynomial and G={g1,...,gk} a set of polynomials in r,
10  * returns a matrix Q=(q1,...,qk) over r such that
11  * f = q1*g1+...+qk*gk+s
12  * is a determinate division with remainder s.
13  */
14 matrix divisionDiscardingRemainder(const poly f, const ideal G, const ring r);
15 
16 /**
17  * Computes a division discarding remainder of F with respect to G.
18  * Given F={f1,...,fl} and G={g1,...,gk} two sets of polynomials in r,
19  * returns a matrix Q=(qij) i=1,..,k j=1,...,l over r such that
20  * fj = q1j*g1+...+qkj*gk+sj
21  * is a determinate division with remainder sj for all j=1,...,l.
22  */
23 matrix divisionDiscardingRemainder(const ideal F, const ideal G, const ring r);
24 
25 /**
26  * Let w be the uppermost weight vector in the matrix defining the ordering on r.
27  * Let I be a Groebner basis of an ideal in r, inI its initial form with respect w.
28  * Given an w-homogeneous element m of inI, computes a witness g of m in I,
29  * i.e. g in I such that in_w(g)=m.
30  */
31 poly witness(const poly m, const ideal I, const ideal inI, const ring r);
32 
33 /**
34  * Computes witnesses in J for inI
35  * Given inI={h1,...,hl} and J={g1,...,gk} two sets of polynomials in r,
36  * returns a set I={f1,...,fl} of <g1,...,gk> such that
37  * in_w(fj)=hj for all j=1,...,l,
38  * where w denotes the uppoermost weight vector in the matrix defining the ordering on r.
39  * Assumes that hj is an element of <in_w(g1),...,in_w(gk)>
40  */
41 ideal witness(const ideal inI, const ideal J, const ring r);
42 
43 #endif
int m
Definition: cfEzgcd.cc:121
FILE * f
Definition: checklibs.c:9
STATIC_VAR TreeM * G
Definition: janet.cc:31
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
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