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abstract: 'In this paper we present some extensions of the Ailon-Rudnick Theorem, which says that if $f,g\in\C[T]$, then $\gcd(f^n-1,g^m-1)$ is bounded for all $n,m\ge 1$. More precisely, using a uniform bound for the number of torsion points on curves and results on the intersection of curves with algebraic subgroups of codimension at least $2$, we present two such extensions in the univariate case. We also give two multivariate analogues of the Ailon-Rudnick Theorem based on Hilbert’s irreducibility theorem and a result of Granville and Rudnick about torsion points on hypersurfaces.'
address: 'School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia'
author:
- Alina Ostafe
title: 'On some extensions of the Ailon-Rudnick Theorem'
---

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Introduction
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Motivation
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Let $a,b$ be multiplicatively independent positive integers and $\varepsilon>0$. Bugeaud, Corvaja and Zannier [@BCZ] have proved that $$\gcd\(a^n-1,b^n-1\)\le \exp(\varepsilon n)$$ as $n$ tends to infinity. Corvaja and Zannier [@CZ05] have generalised this result and replaced $a^n,b^n$ with multiplicatively independent $S$-units $u,v\in\Z$.

In the function field case, Ailon and Rudnick [@AR Theorem 1] proved that 