Enhancing the Extended Hensel Construction by Using Gröbner Bases

Abstract

Contrary to that the general Hensel construction (GHC) uses univariate initial Hensel factors, the extended Hensel construction (EHC) uses multivariate initial Hensel factors determined by the Newton polygon of the given multivariate polynomial. In the EHC so far, Moses-Yun’s (MY) interpolation functions (see the text) are used for Hensel lifting, but the MY functions often become huge when the degree w.r.t. the main variable is large. In this paper, we propose an algorithm which uses, instead of MY functions, Gröbner bases of two initial factors which are homogeneous w.r.t. the main variable and the total-degree variable for sub-variables. The Hensel factors computed by the EHC are polynomials in the main variable with coefficients of mostly rational functions in sub-variables. We propose a method which converts the rational functions into polynomials by replacing the denominators by system variables. Each of the denominators is determined by the lowest order element of a Gröbner basis. Preliminary experiments show that our new EHC method is much faster than the previous one.