Factorization of bivariate sparse polynomials

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.Date: 12/9/2018.

Higher connectivity of tropicalizations

“…The key idea for the proof of Theorem 5 is the following subtorus analogue of Bertini's Theorem due to [9,24]. We also make crucial use of some modifications due to [1]. Following [1] we will say that a map π : X → (K * ) d satisfies the property PB (pullback) if the pullback λ * X := X × (K * ) d (K * ) d of X along π and λ is irreducible for every isogeny λ of (K * ) d .…”

“…We also make crucial use of some modifications due to [1]. Following [1] we will say that a map π : X → (K * ) d satisfies the property PB (pullback) if the pullback λ * X := X × (K * ) d (K * ) d of X along π and λ is irreducible for every isogeny λ of (K * ) d . An isogeny of (K * ) d is a surjective group homomorphism with finite kernel, so can be represented by a d × d rank d matrix with integer entries.…”

What Can (and Can't) we Do with Sparse Polynomials?

“…Specifically, when analyzing algorithms for dense polynomials, the most important measure is the degree bound D ∈ N such that deg f < D. The size of the dense representation of a univariate polynomial is D ring elements, and many operations can be performed in D O (1) ring operations, or even D(log D) O (1) .…”

“…For simplicity of presentation, and because the algorithmic work with sparse polynomials is still at a much more coarse level than, say, that of integers and dense polynomials, we frequently use the soft-oh notation O(γ 1) , where γ is some running-time function.…”

“…Finding high-degree sparse factors remains a challenge in almost all cases. Very recent work by [1] proves that essentially all bivariate high-degree factors of a bivariate rational polynomial must be sparse, which provides some new hope that this problem is tractable.…”