Sur le d\'eveloppement en fraction continue d'une g\'en\'eralisation de la cubique de Baum et Sweet

TL;DR

This paper explores the continued fraction expansions of algebraic power series solutions to generalized equations of the form TX^{r+1}+X-T=0 over finite fields, extending previous work on the Baum--Sweet series.

Contribution

It generalizes the known continued fraction analysis from the Baum--Sweet series to a broader class of algebraic power series defined by equations with prime power exponents.

Findings

Describes continued fraction expansions for solutions of TX^{r+1}+X-T=0

Extends previous results from the Baum--Sweet series to more general equations

Provides a method to analyze algebraic power series over finite fields

Abstract

In 1976, Baum and Sweet gave the first example of a power series that is algebraic over the field $\mathbb F_2(T)$ and whose continued fraction expansion has partial quotients with bounded degree. This power series is the unique solution of the equation $TX^3+X-T=0$. In 1986, Mills and Robbins described an algorithm that allows to compute the continued fraction expansion of the Baum--Sweet power series. In this paper, we consider the more general equations $TX^{r+1}+X-T=0$, where $r$ is a power of a prime number $p$. Such an equation has a unique solution in the field $\mathbb F_p((T^{-1}))$. Applying an approach already used by Lasjaunias, we give a description of the continued fraction expansion of these algebraic power series.