Resolving zero-divisors using Hensel lifting

TL;DR

This paper introduces a Hensel lifting-based algorithm to handle zero-divisors in modular computations over triangular sets, specifically improving GCD calculations of univariate polynomials with rational coefficients.

Contribution

It presents a novel modular algorithm leveraging Hensel lifting to effectively manage zero-divisors, extending algebraic number theory techniques to polynomial GCD computations.

Findings

Algorithm successfully handles zero-divisors in modular GCD computations.

Implementation in Maple shows competitive performance.

Generalizes previous algebraic number theory methods.

Abstract

Algorithms which compute modulo triangular sets must respect the presence of zero-divisors. We present Hensel lifting as a tool for dealing with them. We give an application: a modular algorithm for computing GCDs of univariate polynomials with coefficients modulo a radical triangular set over the rationals. Our modular algorithm naturally generalizes previous work from algebraic number theory. We have implemented our algorithm using Maple's RECDEN package. We compare our implementation with the procedure RegularGcd in the RegularChains package.