A note on hyperquadratic continued fractions in characteristic 2 with partial quotients of degree 1

TL;DR

This paper explores a specific family of algebraic, nonquadratic power series over finite fields of characteristic 2, with continued fractions having degree-one partial quotients, highlighting connections with similar families in odd characteristic.

Contribution

It introduces a new family of algebraic power series in characteristic 2 with degree-one partial quotients, linking them to known families in odd characteristic.

Findings

Identified a family of algebraic, nonquadratic power series with degree-one partial quotients in characteristic 2.

Established a connection between these series and similar families in odd characteristic.

Provided insights into the structure of continued fractions over finite fields of characteristic 2.

Abstract

In this note, we describe a family of particular algebraic, and nonquadratic, power series over an arbitrary finite field of characteristic 2, having a continued fraction expansion with all partial quotients of degree one. The main purpose being to point out a common origin with another analogue family in odd characteristic, which has been studied in [LY1].