# Composite rational functions expressible with few terms

• Volume: 014, Issue: 1, page 175-208
• ISSN: 1435-9855

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## Abstract

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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 $f\left(x\right)=g\left(h\left(x\right)\right)$, 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 $h\left(x\right)$ 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.

## How to cite

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Fuchs, Clemens, and Zannier, Umberto. "Composite rational functions expressible with few terms." Journal of the European Mathematical Society 014.1 (2012): 175-208. <http://eudml.org/doc/277725>.

@article{Fuchs2012,
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.},
author = {Fuchs, Clemens, Zannier, Umberto},
journal = {Journal of the European Mathematical Society},
keywords = {rational functions; decomposability; lacunarity; Brownawell–Masser inequality; Puiseux expansion; permutation groups with cyclic two-orbit; rational functions; decomposability; lacunarity; Brownawell-Masser inequality.},
language = {eng},
number = {1},
pages = {175-208},
publisher = {European Mathematical Society Publishing House},
title = {Composite rational functions expressible with few terms},
url = {http://eudml.org/doc/277725},
volume = {014},
year = {2012},
}

TY - JOUR
AU - Fuchs, Clemens
AU - Zannier, Umberto
TI - Composite rational functions expressible with few terms
JO - Journal of the European Mathematical Society
PY - 2012
PB - European Mathematical Society Publishing House
VL - 014
IS - 1
SP - 175
EP - 208
AB - 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.
LA - eng
KW - rational functions; decomposability; lacunarity; Brownawell–Masser inequality; Puiseux expansion; permutation groups with cyclic two-orbit; rational functions; decomposability; lacunarity; Brownawell-Masser inequality.
UR - http://eudml.org/doc/277725
ER -

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