The Thue-Morse continued fractions in characteristic $2$ are algebraic

TL;DR

This paper proves that certain power series over the field with two elements, constructed from the Thue-Morse sequence, are algebraic of degree four, revealing new algebraic properties in characteristic 2.

Contribution

It demonstrates that power series with Thue-Morse partial quotients over ${f F}_2$ are algebraic of degree four, a novel result in the context of characteristic 2.

Findings

$\xi_{a,b}$ is algebraic of degree 4

Thue-Morse sequence leads to algebraic power series

New insights into algebraic properties in characteristic 2

Abstract

Let $a, b$ be distinct, non-constant polynomials in ${\mathbb F}_2 [z]$. Let $\xi_{a, b}$ be the power series in ${\mathbb F}_2((z^{-1}))$ whose sequence of partial quotients is the Thue-Morse sequence over $\{a, b\}$. We establish that $\xi_{a,b}$ is algebraic of degree $4$.