On the periodicity problem for the continued fraction expansion of elements of hyperelliptic fields with fundamental $S$-units of degree at most 11

We solve the problem of describing square-free polynomials $f(x)\in k[x]$ with a periodic expansion of $\sqrt{f(x)}$ into a functional continued fraction in $k((x))$, where $k$ is a number field and the degree of the corresponding fundamental $S$-unit of the hyperelliptic field $k(x)(\sqrt{f(x)})$ is less than or equal to 11.