fglm -- convert a Groebner basis

Synopsis

Usage:

H = fglm(G, R)

H = fglm(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
Outputs:
- H, a Gröbner basis, the new Groebner basis in the target monomial order

Description

FGLM 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 : R1 = QQ[x,y,z];

i2 : I1 = ideal(x^2 + 2*y^2 - y - 2*z, x^2 - 8*y^2 + 10*z - 1, x^2 - 7*y*z);

o2 : Ideal of R1

i3 : R2 = QQ[x,y,z, MonomialOrder => Lex];

i4 : fglm(I1, R2)

o4 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]

o4 : GroebnerBasis

Caveat

The ideal I generated by G must be zero-dimensional. 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 fglm :

fglm(GroebnerBasis,Ring)
fglm(Ideal,Ring)

For the programmer