Bounded-degree factors of lacunary multivariate polynomials

TL;DR

This paper introduces a new polynomial-time algorithm for computing bounded-degree factors of lacunary multivariate polynomials over number fields, utilizing a gap theorem and Newton polytope analysis.

Contribution

The paper presents a novel algorithm that efficiently factors lacunary multivariate polynomials by reducing the problem to univariate and low-degree cases using elementary exponent vector manipulations.

Findings

Algorithm works in polynomial time relative to lacunary size and degree bound

Applicable to any field of zero characteristic

Uses Newton polytope and Puiseux series for correctness and complexity analysis

Abstract

In this paper, we present a new method for computing bounded-degree factors of lacunary multivariate polynomials. In particular for polynomials over number fields, we give a new algorithm that takes as input a multivariate polynomial f in lacunary representation and a degree bound d and computes the irreducible factors of degree at most d of f in time polynomial in the lacunary size of f and in d. Our algorithm, which is valid for any field of zero characteristic, is based on a new gap theorem that enables reducing the problem to several instances of (a) the univariate case and (b) low-degree multivariate factorization. The reduction algorithms we propose are elementary in that they only manipulate the exponent vectors of the input polynomial. The proof of correctness and the complexity bounds rely on the Newton polytope of the polynomial, where the underlying valued field consists of Puiseux series in a single variable.