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Let $g(x)$ be a fixed non-constant complex polynomial. It was conjectured bySchinzel that if $g(h(x))$ has boundedly many terms, then $h(x)\in \C[x]$ mustalso have boundedly many terms. Solving an older conjecture raised by R\'enyiand 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 fullSchinzel's conjecture (actually in sharper form) by a completely differentmethod. Simultaneously we establish an "algorithmic" parametric description ofthe general decomposition $f(x)=g(h(x))$, where $f$ is a polynomial with agiven number of terms and $g,h$ are arbitrary polynomials. As a corollary, thisimplies for instance that a polynomial with $l$ terms and given coefficients isnon-trivially decomposable if and only if the degree-vector lies in the unionof certain finitely many subgroups of $\Z^l$.

Composite lacunary polynomials and the proof of a conjecture of Schinzel