# grobner basis of an ideal dependent on some parameter

Suppose $$I = \langle f_1, ... , f_l \rangle$$ is an ideal generated by polynomials $$f \in k[x_1,\dots,x_n]$$, where $$k$$ is a field of rational functions in some parameters $$s_1,\dots,s_m$$.

What are the techniques to study the dependence of the grobner basis of $$I$$ (with respect to some order) on the parameters $$s$$? Is it possible to find "regions" in the parameter space where the basis is doesn't change?

• The relevant notion is that of a "Groebner system," closely related to comprehensive Groebner bases introduced by Weispfenning. The regions of the parameter space are constructible sets in general. One generally expects these calculations to be immense, though I believe implementations do exist. – tim Feb 21 at 22:45