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# Algebra Seminar

Wednesday, October 28, 2020 — 2:00 PM EDT

Seda Albayrak, Department of Pure Mathematics, University of Waterloo

"A refinement of Christol’s theorem"

Christol's theorem is one of the fundamental results in the theory of finite-state automata. It says that a formal power series $F(x)=\sum_n a_n x^n$ with coefficients in a finite field $\mathbb{F}_q$, $q$ a power of a prime $p$, is algebraic over the field of rational functions $\mathbb{F}_q(x)$ if and only if the sequence $\{a_n\}$ is $p$-automatic. The support of an algebraic power series, i.e.the set of $n$ for which $a_n\neq 0$, is an automatic subset of $\mathbb{N}$. There is a dichotomy for automatic sets that says automatic sets are either sparse, having at most ${\rm O}((\log \, n)^d)$ elements of size at most $n$ for some $d\ge 1$ and all $n$; or they are non-sparse, have at least $n^{\alpha}$ elements of size at most $n$ for some positive number $\alpha$ and all $n$ sufficiently large. In a joint work with Jason Bell, we characterize algebraic power series with sparse support, giving a refinement of Christol’s theorem. In fact we are able to prove our result in a more general setting, that is for generalized power series, studied and characterized by Kedlaya.

### August 2021

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