CUBIC FORMAL POWER SERIES IN CHARACTERISTIC 2 WITH UNBOUNDED PARTIAL QUOTIENTS

Abstract

There is a theory of continued fractions for formal power series in $x^{−1}$ with coefficients in a field $\mathbb{F}_q$. This theory bears a close analogy with classical continued fractions for real numbers with formal power 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. We give a family of cubic power series over $\mathbb{F}_2$ with unbounded partial quotients. To be more precise, let $f \in \mathbb{F}_2((x^{−1}))$ such that $f$ is not polynomial but $f^3$ is polynomial with degreed $d \in 3\mathbb{N}$, we prove that the continued fraction expansion of $f$ is unbounded.