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

Let $a, b$ be distinct, non-constant polynomials in ${\mathbb F}_2 [z]$. Let $ξ_{a, b}$ 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 $ξ_{a,b}$ is algebraic of degree $4$.

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