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A Skolem–Mahler–Lech theorem in positive characteristic and finite automata

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Abstract

Lech proved in 1953 that the set of zeroes of a linear recurrence sequence in a field of characteristic 0 is the union of a finite set and finitely many infinite arithmetic progressions. This result is known as the Skolem–Mahler–Lech theorem. Lech gave a counterexample to a similar statement in positive characteristic. We will present some more pathological examples. We will state and prove a correct analog of the Skolem–Mahler–Lech theorem in positive characteristic. The zeroes of a recurrence sequence in positive characteristic can be described using finite automata.

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  1. Department of Mathematics, University of Michigan, Ann Arbor, USA

    Harm Derksen

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  1. Harm Derksen
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Derksen, H. A Skolem–Mahler–Lech theorem in positive characteristic and finite automata. Invent. math. 168, 175–224 (2007). https://doi.org/10.1007/s00222-006-0031-0

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  • DOI: https://doi.org/10.1007/s00222-006-0031-0

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