Modular Las Vegas Algorithms for Polynomial Absolute Factorization

TL;DR

This paper introduces a probabilistic algorithm for absolute irreducibility testing and polynomial factorization over integers, leveraging properties of Newton polytopes and lattice reduction, with promising results for high-degree polynomials.

Contribution

It presents a novel Las Vegas algorithm for polynomial irreducibility testing and factorization that extends to multivariate cases, improving efficiency for high-degree polynomials.

Findings

Effective for polynomials up to degree 400

Utilizes properties of Newton polytopes and prime selection

Shows promising computational results in Maple

Abstract

Let $f(X,Y) \in \ZZ[X,Y]$ be an irreducible polynomial over $\QQ$. We give a Las Vegas absolute irreducibility test based on a property of the Newton polytope of $f$, or more precisely, of $f$ modulo some prime integer $p$. The same idea of choosing a $p$ satisfying some prescribed properties together with $LLL$ is used to provide a new strategy for absolute factorization of $f(X,Y)$. We present our approach in the bivariate case but the techniques extend to the multivariate case. Maple computations show that it is efficient and promising as we are able to factorize some polynomials of degree up to 400.