A generalised Skolem-Mahler-Lech theorem for affine varieties

Abstract

The Skolem-Mahler-Lech theorem states that if $f(n)$ is a sequence given by a linear recurrence over a field of characteristic 0,then the set of $m$ such that $f(m)$ is equal to 0 is the union of a finite number of arithmetic progressions in $m\ge 0$ and a finite set. We prove that if $X$ is a subvariety of an affine variety $Y$ over a field of characteristic 0 and ${\bf q}$ is a point in $Y$, and $σ$ is an automorphism of $Y$, then the set of $m$ such that $σ^m({\bf q})$ lies in $X$ is a union of a finite number of complete doubly-infinite arithmetic progressions and a finite set. We show that this is a generalization of the Skolem-Mahler-Lech theorem.