Explicit bounds for composite lacunary polynomials

Abstract

Let $f, g, h\in \mathbb{C}\left[x\right]$ be non-constant complex polynomials satisfying $f(x)=g(h(x))$ and let $f$ be lacunary in the sense that it has at most $l$ non-constant terms. Zannier proved that there exists a function $B_1(l)$ on $\mathbb{N}$, depending only on $l$ and with the property that $h(x)$ can be written as the ratio of two polynomials having each at most $B_1(l)$ terms. Here, we give explicit estimates for this function or, more precicely, we prove that one may take for instance \[B_1(l)=(4l)^{(2l)^{(3l)^{l+1}}}.\] Moreover, in the case $l=2$, a better result is obtained using the same strategy.