Algebraic continued fractions in F_q((T^{-1})) and recurrent sequences in F_q

TL;DR

This paper explores the continued fraction expansions of hyperquadratic algebraic power series over finite fields, revealing explicit forms and recurrent sequences, especially in odd characteristic and non-prime fields.

Contribution

It provides explicit continued fraction expansions for a broad family of hyperquadratic power series in odd characteristic, linking algebraic series to recurrent sequences over finite fields.

Findings

Explicit continued fraction expansions for hyperquadratic series in odd characteristic

Identification of recurrent sequences in finite fields beyond prime fields

Comparison of algebraic power series with quadratic real numbers

Abstract

There exists a particular subset of algebraic power series over a finite field which, for different reasons, can be compared to the subset of quadratic real numbers. The continued fraction expansion for these elements, called hyperquadratic, can sometimes be made explicit. Here we describe this expansion for a wide family of hyperquadratic power series in odd characteristic. This leads to consider interesting recurrent sequences in the finite base field when it is not a prime field.