On some extensions of the Ailon-Rudnick Theorem

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.

Theorem 1.3 of vers. 2 is generalised

References in corpus (1)

• The density of prime divisors in the arithmetic dynamics of quadratic polynomials