On composite lacunary polynomials and the proof of a conjecture of Schinzel

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)∈ℂ[x] must also have boundedly many terms. Solving an older conjecture raised by Rényi and by Erdös, 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 a suitable element of the Boolean algebra generated by the subgroups of ℤ^l.