Factoring bivariate lacunary polynomials without heights

TL;DR

This paper introduces a new polynomial-time algorithm for factoring bivariate lacunary polynomials, leveraging a novel Gap Theorem that simplifies zero-testing of specific polynomial forms without relying on coefficient heights.

Contribution

It presents an elementary, height-independent algorithm for multilinear factorization of bivariate lacunary polynomials based on a new Gap Theorem.

Findings

Algorithm efficiently finds multilinear factors

Applicable over fields with large finite characteristic

Operates in probabilistic polynomial time

Abstract

We present an algorithm which computes the multilinear factors of bivariate lacunary polynomials. It is based on a new Gap Theorem which allows to test whether a polynomial of the form P(X,X+1) is identically zero in time polynomial in the number of terms of P(X,Y). The algorithm we obtain is more elementary than the one by Kaltofen and Koiran (ISSAC'05) since it relies on the valuation of polynomials of the previous form instead of the height of the coefficients. As a result, it can be used to find some linear factors of bivariate lacunary polynomials over a field of large finite characteristic in probabilistic polynomial time.