Contact Info
Pure MathematicsUniversity of Waterloo
200 University Avenue West
Waterloo, Ontario, Canada
N2L 3G1
Departmental office: MC 5304
Phone: 519 888 4567 x43484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca
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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 finitestate 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 nonsparse, 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
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Departmental office: MC 5304
Phone: 519 888 4567 x43484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca
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