On some extensions of the Ailon-Rudnick Theorem

TL;DR

This paper extends the Ailon-Rudnick Theorem by providing new bounds and multivariate analogues using advanced algebraic geometry tools like torsion points, curve intersections, and Hilbert's irreducibility theorem.

Contribution

It introduces novel univariate and multivariate extensions of the Ailon-Rudnick Theorem employing bounds on torsion points and algebraic subgroup intersections.

Findings

Boundedness of gcd for polynomial powers in univariate case

Two multivariate analogues based on Hilbert's irreducibility theorem

Use of torsion point results on curves and hypersurfaces

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.