New Results on the Periodicity Problem for Continued Fractions of Elements of Hyperelliptic Fields

We study the problem of describing square-free polynomials $f(x)$ of odd degree with periodic expansion of $\sqrt {f(x)}$ into a functional continued fraction in $k((x))$, where $k\subseteq \overline {\mathbb Q}$. We obtain a complete description of such polynomials $f(x)$ that does not depend on the field $k$ and the degree of a polynomial, provided that the degree $U$ of the fundamental $S$-unit of the corresponding hyperelliptic field $k(x)(\sqrt {f(x)})$ either does not exceed $12$ or is even and does not exceed $20$.