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SUMMARY:Seda Albayrak (U of Calgary)
DTSTART:20241213T220000Z
DTEND:20241213T230000Z
DTSTAMP:20260423T023047Z
UID:NT-UBC/8
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NT-UBC/8/">M
 ultivariate generalization of Christol's Theorem</a>\nby Seda Albayrak (U 
 of Calgary) as part of UBC Number theory seminar\n\nLecture held in ESB413
 3.\n\nAbstract\nChristol's theorem (1979)\, which sets ground for many int
 eractions\nbetween theoretical computer science and number theory\, charac
 terizes the\ncoefficients of a formal power series over a finite field of 
 positive\ncharacteristic $p>0$ that satisfy an algebraic equation to be th
 e sequences\nthat can be generated by finite automata\, that is\, a finite
 -state machine takes\nthe base-$p$ expansion of $n$ for each coefficient a
 nd gives the coefficient\nitself as output.  Namely\, a formal power serie
 s $\\sum_{n\\ge 0} f(n) t^n$ over\n$\\mathbb{F}_p$ is algebraic over $\\ma
 thbb{F}_p(t)$ if and only if $f(n)$ is a\n$p$-automatic sequence. However\
 , this characterization does not give the full\nalgebraic closure of $\\ma
 thbb{F}_p(t)$. Later it was shown by Kedlaya (2006)\nthat a description of
  the complete algebraic closure of $\\mathbb{F}_p(t)$ can\nbe given in ter
 ms of $p$-quasi-automatic generalized (Laurent) series. In fact\,\nthe alg
 ebraic closure of $\\mathbb{F}_p(t)$ is precisely generalized Laurent\nser
 ies that are $p$-quasi-automatic. We will characterize elements in the\nal
 gebraic closure of function fields over a field of positive characteristic
 \nvia finite automata in the multivariate setting\, extending Kedlaya's re
 sults.\nIn particular\, our aim is to give a description of the \\textit{f
 ull algebraic\nclosure} for \\textit{multivariate} fraction fields of posi
 tive characteristic.\n
LOCATION:https://researchseminars.org/talk/NT-UBC/8/
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