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

• 成果类型: Article
• 署名作者: Derksen, Harm
• 署名单位: University of Michigan System; University of Michigan
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-006-0031-0
• 发表日期: 2007
• 页码: 175-224
• 关键词: linear recurrence sequences; EQUATIONS

摘要

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.