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.
13-09-23

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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.