Quartic Power Series in $\f_3((t^{-1}))$

TL;DR

This paper explores special algebraic power series over the finite field _3, generalizing previous results on continued fractions with degree 1 partial quotients and proposing a conjecture for broader cases.

Contribution

It generalizes Lasjaunias's earlier work on power series with degree 1 partial quotients and introduces a new conjecture about their structure.

Findings

Extended the class of algebraic power series with degree 1 partial quotients

Proposed a conjecture on the structure of such power series with initial exceptions

Built upon previous results to deepen understanding of continued fractions over finite fields

Abstract

We are concerned with power series in 1/T over a finite field of 3 elements $\F_3$. In a previous article, Alain Lasjaunias investigated the existence of particular power series of elements algebraic over $\F_3[T]$, having all partial quotients of degree 1 in their continued fraction expansion. Here, we generalize his result and we make a conjecture about the elements with all partial quotients of degree 1, except maybe the first ones.