## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x33484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

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Please note: The University of Waterloo is closed for all events until further notice.

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.

Zoom meeting: https://us02web.zoom.us/j/81125421802?pwd=c3NaZmNRVnJKMkk0U0hLZXpVNTBtQT09

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x33484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land promised to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Indigenous Initiatives Office.