On some extensions of the Ailon-Rudnick theorem

In this paper we present some extensions of the Ailon-Rudnick theorem, which says that if ƒ, g ∈ C[T ], then gcd(ƒ ⁿ - 1, g^m - 1) is bounded for all n, m >= 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.