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

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 $4$.

arXiv comment21 pages