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

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.