A lifting and recombination algorithm for rational factorization of sparse polynomials

TL;DR

This paper introduces a novel lifting and recombination algorithm for rational bivariate polynomial factorization that leverages Newton polytope geometry, achieving polynomial complexity based on the polytope's volume.

Contribution

It presents a deterministic, geometry-based algorithm for sparse polynomial factorization, extending previous methods with improved complexity analysis.

Findings

Algorithm has polynomial complexity in the Newton polytope volume.

Utilizes algebraic osculation criteria in toric varieties.

Provides a sparse version of Lecerf's factorization algorithm.

Abstract

We propose a new lifting and recombination scheme for rational bivariate polynomial factorization that takes advantage of the Newton polytope geometry. We obtain a deterministic algorithm that can be seen as a sparse version of an algorithm of Lecerf, with now a polynomial complexity in the volume of the Newton polytope. We adopt a geometrical point of view, the main tool being derived from some algebraic osculation criterions in toric varieties.