FGLM :: fglm

# fglm -- convert a Groebner basis

## Synopsis

• Usage:
H = fglm(G, R)
H = fglm(I, R)
• Inputs:
• G, , the starting Groebner basis
• I, an ideal, the starting ideal
• R, a ring, a ring with the target monomial order
• Outputs:
• H, , 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.