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A Skolem-Mahler-Lech Theorem in Positive Characteristic and Finite Automata

Harm Derksen
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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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Harm Derksen. 2005-10-27. A Skolem-Mahler-Lech Theorem in Positive Characteristic and Finite Automata. https://arxiv.org/abs/math/0510583

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