SFU Number Theory and Algebraic Geometry Seminar: Seda Albayrak

Christol's theorem (1979), which sets ground for many interactions between theoretical computer science and number theory, characterizes the coefficients of a formal power series over a finite field of positive characteristic $p>0$ that satisfy an algebraic equation to be the sequences that can be generated by finite automata, that is, a finite-state machine takes the base-$p$ expansion of $n$ for each coefficient and gives the coefficient itself as output. Namely, a formal power series $\sum_{n\ge 0} f(n) t^n$ over $\mathbb{F}_p$ is algebraic over $\mathbb{F}_p (t)$ if and only if $f(n)$ is a $p$-automatic sequence. However, this characterization does not give the full algebraic closure of $\mathbb{F}_p (t)$. Later it was shown by Kedlaya (2006) that a description of the complete algebraic closure of $\mathbb{F}_p (t)$ can be given in terms of $p$-quasi-automatic generalized (Laurent) series. In fact, the algebraic closure of $\mathbb{F}_p (t)$ is precisely generalized Laurent series that are $p$-quasi-automatic. We will characterize elements in the algebraic closure of function fields over a field of positive characteristic via finite automata in the multivariate setting, extending Kedlaya's results. In particular, our aim is to give a description of the full algebraic closure for multivariate fraction fields of positive characteristic.