A generalized Skolem-Mahler-Lech theorem for affine varieties

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 generalization of the Skolem-Mahler-Lech theorem. key words: automorphisms, affine space, Skolem-Mahler-Lech theorem Primary classification: 11D45 Secondary classification: 14R10, 11Y55, 11D88