Explicit bounds for composite lacunary polynomials

TL;DR

This paper provides explicit bounds on the number of terms in polynomial decompositions involving lacunary polynomials, improving understanding of their structure and offering concrete estimates for the complexity of such decompositions.

Contribution

The paper derives explicit, computable bounds for the number of terms in polynomial decompositions involving lacunary polynomials, extending prior qualitative results.

Findings

Explicit bound B_1(l) = (4l)^{(2l)^{(3l)^{l+1}}} for the number of terms in polynomial decompositions.

Improved bounds for the case l=2 using the same strategy.

Enhanced understanding of the structure of lacunary polynomial compositions.

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.