Composite lacunary polynomials and the proof of a conjecture of Schinzel

TL;DR

This paper proves Schinzel's conjecture that polynomials with a bounded number of terms in their composition imply the inner polynomial also has boundedly many terms, using a new method and providing an algorithmic decomposition description.

Contribution

It offers a complete proof of Schinzel's conjecture in a sharper form and introduces an algorithmic approach to polynomial decomposition.

Findings

Proof of Schinzel's conjecture for polynomials with bounded terms

Algorithmic description of polynomial decompositions

Characterization of decomposable polynomials via degree-vector subgroups

Abstract

Let $g(x)$ be a fixed non-constant complex polynomial. It was conjectured by Schinzel that if $g(h(x))$ has boundedly many terms, then $h(x)\in \C[x]$ must also have boundedly many terms. Solving an older conjecture raised by R\'enyi and by Erd\"os, Schinzel had proved this in the special cases $g(x)=x^d$; however that method does not extend to the general case. Here we prove the full Schinzel's conjecture (actually in sharper form) by a completely different method. Simultaneously we establish an "algorithmic" parametric description of the general decomposition $f(x)=g(h(x))$, where $f$ is a polynomial with a given number of terms and $g,h$ are arbitrary polynomials. As a corollary, this implies for instance that a polynomial with $l$ terms and given coefficients is non-trivially decomposable if and only if the degree-vector lies in the union of certain finitely many subgroups of $\Z^l$.