Factorization of bivariate sparse polynomials

TL;DR

This paper proves a function field analogue of Schinzel's conjecture, establishing that sparse bivariate Laurent polynomials have finitely many irreducible factorizations with fixed coefficients, and that these factors are also sparse.

Contribution

It introduces a new finiteness theorem for irreducible factorizations of sparse bivariate Laurent polynomials over complex coefficients.

Findings

Finiteness of irreducible factorizations in polynomial families

Truly bivariate irreducible factors are also sparse

Application of a variant of toric Bertini's theorem

Abstract

We prove a function field analogue of a conjecture of Schinzel on the factorization of univariate polynomials over the rationals. We derive from it a finiteness theorem for the irreducible factorizations of the bivariate Laurent polynomials in families with fixed set of complex coefficients and varying exponents. Roughly speaking, this result shows that the truly bivariate irreducible factors of these sparse Laurent polynomials, are also sparse. The proofs are based on a variant of the toric Bertini's theorem due to Zannier and Fuchs, Mantova and Zannier.