Factorization of bivariate sparse polynomials

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 generalization of a toric Bertini's theorem due to Fuchs, Mantova and Zannier.