Composite rational functions expressible with few terms

Abstract

We consider a rational function $f$ which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number $\ell$ of terms. Then we look at the possible decompositions $f(x)=g(h(x))$, where $g,h$ are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of $g$ is bounded only in terms of $\ell$ (and we provide explicit bounds). This supports and quantifies the intuitive expectation that rational operations of large degree tend to destroy lacunarity. As an application in the context of algebraic dynamics, we show that the minimum number of terms necessary to express an iterate $h^{on}$ of a rational function $h$ tends to infinity with $n$, provided $h(x)$ is not of an explicitly described special shape. The conclusions extend some previous results for the case when $f$ is a Laurent polynomial; the proofs present several features which have not appeared at all in the special cases treated so far.