groebnerWalk -- convert a Groebner basis

Synopsis

Usage:

H = groebnerWalk(G, R)

H = groebnerWalk(I, R)
Inputs:
- G, a Gröbner basis, the starting Groebner basis
- I, an ideal, the starting ideal
- R, a ring, a ring with the target monomial order
Optional inputs:
- Strategy => ..., default value Standard, specify the algorithm for groebnerWalk
Outputs:
- H, a Gröbner basis, the new Groebner basis in the target monomial order

Description

The Groebner walk takes a Groebner basis of an ideal with respect to one monomial order and changes it into a Groebner basis of the same ideal over a different monomial order. The initial order is given by the ring of G and the target order is the order in R. When given an ideal I as input a Groebner basis of I in the ring of I is initially computed directly, and then this Groebner basis is converted into a Groebner basis in the ring R.

i1 : KK = ZZ/32003;

i2 : R1 = KK[x,y,z,u,v, MonomialOrder=>Eliminate 3];

i3 : I1 = ideal(3 - 2*u + 2*u^2 - 2*u^3 - v + u*v + 2*u^2*v^3 - x,
                6*u + 5*u^2 - u^3 + v + u*v + v^2 - y,
                -2 + 3*u - u*v + 2*u*v^2 - z);

o3 : Ideal of R1

i4 : R2 = KK[x,y,z,u,v, MonomialOrder=>Weights=>{0,0,0,1,1}];

i5 : groebnerWalk(I1, R2)

o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]

o5 : GroebnerBasis

Caveat

The target ring R must be the same ring as the ring of G or I, except with different monomial order. R must be a polynomial ring over a field.

Ways to use `groebnerWalk` :

"groebnerWalk(GroebnerBasis,Ring)"
"groebnerWalk(Ideal,Ring)"

For the programmer

The object groebnerWalk is a method function with options.