Mathematics Colloquium Date: Wednesday, 13 September 2023 Time: 3:30  4:30 pm Venue: Seminar Hall On algebraicity of power series in positive characteristic Xavier Caruso University of Bordeaux. 130923 Abstract Let $f(x) = \sum_{n \geq 0} a_n x^n$ be a power series with coefficients in a field $k$. In general, recognizing whether $f(x)$ is algebraic over $k(x)$, i.e. whether there exists a bivariate polynomial $P(x,y)$ such that $P(x, f(x))$ is a difficult question. However, when $k$ has positive characteristic, there is a beautiful criterion, known as Christol's theorem, allowing or reading algebraicity on the sequence $(a_n)_{n \geq 0}$ of the coefficients. In this talk, I will discuss Christol's theorem, illustrate it with many examples and give several applications. I will also investigate how this could help to attack the case of characteristic zero.
