Various Enhancements for Extended Hensel Construction of Sparse Multivariate Polynomials

The extended Hensel construction (EHC) is a direct extension of the generalized Hensel construction (GHC), and it targets sparse multivariate polynomials for which the GHC breaks down. The EHC consists of two Hensel constructions which we call separation of "maximal" and "minimal" Hensel factors (see the text). As for the minimal Hensel factor separation, very recently, we enhanced the old algorithm largely by using Groebner basis of two initial factors and syzygies for the elements of the basis. In this paper, we first improve the old algorithm for maximal Hensel factors. We then enhance further the Groebner basis computation in our recent algorithm. The latter is based on a theoretical analysis of the Groebner bases. Simple experiments show that the improved part for the minimal Hensel factors is much faster than the recent one.