Unlikely intersections and multiple roots of sparse polynomials

TL;DR

This paper establishes a structure theorem for the multiple non-cyclotomic irreducible factors of sparse polynomials, showing they are also sparse, and provides bounds related to torsion curve intersections.

Contribution

It introduces a new structure theorem for sparse polynomial factors and extends a theorem of Bombieri and Zannier with explicit bounds.

Findings

Multiple non-cyclotomic irreducible factors are sparse.

Provides explicit bounds on torsion curve intersections.

Supports a conjecture of Bolognesi and Pirola.

Abstract

We present a structure theorem for the multiple non-cyclotomic irreducible factors appearing in the family of all univariate polynomials with a given set of coefficients and varying exponents. Roughly speaking, this result shows that the multiple non-cyclotomic irreducible factors of a sparse polynomial, are also sparse. To prove this, we give a variant of a theorem of Bombieri and Zannier on the intersection of a fixed subvariety of codimension 2 of the multiplicative group with all the torsion curves, with bounds having an explicit dependence on the height of the subvariety. We also use this latter result to give some evidence on a conjecture of Bolognesi and Pirola.