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Presented By: Department of Mathematics

Student Arithmetic

Formal power series, finite automata, and Christol's theorem

Let F be a finite field. The ring of formal power series F[[t]] is the t-adic completion of the polynomial ring F[t]. Completing F[t] introduces many new elements: most are transcendental, but some are algebraic. How do we recognize the formal power series which are algebraic over F[t]? Christol gave a beautifully simple answer to this question using the concept of finite automaton from theoretical computer science. I will discuss the proof of this theorem, give applications and examples, and note extensions as time permits. No background with finite automata or computer science will be assumed. Speaker(s): Trevor Hyde (UM)

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