Cubic Laurent Series in Characteristic 2 with Bounded Partial Quotients

TL;DR

This paper explores continued fraction theory for Laurent series over finite fields of characteristic 2, focusing on bounded partial quotients in solutions of specific cubic equations, and identifies distinct cases with low-degree polynomial coefficients.

Contribution

It classifies cases of Laurent series solutions with bounded partial quotients for cubic equations with low-degree polynomial coefficients over GF(2).

Findings

Three distinct cases identified for degree ≤ 1 polynomial coefficients.

Empirical observations on relations between Laurent series roots of the same cubic.

Analysis of bounded partial quotients in characteristic 2 Laurent series.

Abstract

There is a theory of continued fractions for Laurent series in x^{-1} with coefficients in a field F. This theory bears a close analogy with classical continued fractions for real numbers with Laurent series playing the role of real numbers and the sum of the terms of non-negative degree in x playing the role of the integral part. In this paper we survey the Laurent series u, with coefficients in a finite extension of gf(2), that satisfy an irreducible equation of the form a_0(x)+ a_1(x)u+a_2(x)u^2 + a_3(x)u^3=0 with a_3 \ne 0 and where the a_i are polynomials of low degree in x with coefficients in gf(2). We are particularly interested in the cases in which the sequence of partial quotients is bounded (only finitely many distinct partial quotients occur). We find that there are three essentially different cases when the a_i(x) have degree \le 1. We also make some empirical observations concerning relations between different Laurent series roots of the same cubic.