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Journal of the London Mathematical Society 2006 73(2):367-379; doi:10.1112/S002461070602268X
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© The London Mathematical Society

A Generalised Skolem–Mahler–lech Theorem for Affine Varieties

Jason P. Bell

Department of Mathematics, Simon Fraser University 8888 University Drive, Burnaby, BC V5A 1S6, Canada jpb{at}math.sfu.ca

Received 20 February 2005. Revision received 11 July 2005.

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 {sigma} is an automorphism of Y, then the set of m such that {sigma}m(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.


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