Rational approximations to algebraic Laurent series with coefficients in a finite field

TL;DR

This paper establishes upper bounds for the irrationality exponent of algebraic Laurent series over finite fields, utilizing automata theory and Christol's theorem, and introduces methods to improve these bounds in specific cases.

Contribution

It provides a new approach that significantly refines existing bounds on irrationality exponents for algebraic Laurent series over finite fields.

Findings

Derived a general upper bound for irrationality exponents

Developed a new method to improve bounds in many cases

Computed exact irrationality exponents for specific algebraic Laurent series

Abstract

In this paper we give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field. Our proof is based on a method introduced in a different framework by Adamczewski and Cassaigne. It makes use of automata theory and, in our context, of a classical theorem due to Christol. We then introduce a new approach which allows us to strongly improve this general bound in many cases. As an illustration, we give few examples of algebraic Laurent series for which we are able to compute the exact value of the irrationality exponent.