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 ⩾ 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 q is a point in Y, and σ is an automorphism of Y, then the set of m such that σm(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 generalisation of the Skolem–Mahler–Lech theorem.