Computing Generic Fibers of Polynomial Ideals with FGLM and Hensel Lifting

Abstract

We describe a version of the FGLM algorithm that can be used to compute generic fibers of positive-dimensional polynomial ideals. It combines the FGLM algorithm with a Hensel lifting strategy. In analogy with Hensel lifting, we show that this algorithm has a complexity quasi-linear in the number of terms of certain $\mathfrak{m}$-adic expansions we compute. Some provided experimental data also demonstrates the practical efficacy of our algorithm.