On some extensions of the Ailon-Rudnick Theorem

Alina Ostafe

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arXiv · 1505.03957

On some extensions of the Ailon-Rudnick Theorem

Alina Ostafe

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Abstract

In this paper we present some extensions of the Ailon-Rudnick Theorem, which says that if $f,g\in{\mathbb C}[T]$, 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.

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Alina Ostafe. 2015-07-14. On some extensions of the Ailon-Rudnick Theorem. https://arxiv.org/abs/1505.03957

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