The criterion of periodicity of continued fractions of key elements in hyperelliptic fields

The periodicity and quasi-periodicity of functional continued fractions in the hyperelliptic field $L = \mathbb{Q}(x)(\sqrt {f})$ has a more complex nature, than the periodicity of the numerical continued fractions of the elements of a quadratic fields. It is known that the periodicity of a continued fraction of the element $\sqrt{f}/h^{g + 1}$, constructed by valuation associated with a polynomial $h$ of first degree, is equivalent to the existence of nontrivial $S$-units in a field $L$ of the genus $g$ and is equivalent to the existence nontrivial torsion in a group of classes of divisors. This article has found an exact interval of values of $s \in \mathbb{Z}$ such that the elements $\sqrt {f}/h^s $ have a periodic decomposition into a continued fraction, where $f \in \mathbb{Q}[x] $ is a squarefree polynomial of even degree. For polynomials $f$ of odd degree, the problem of periodicity of continued fractions of elements of the form $\sqrt {f}/h^s $ are discussed in the article [5], and it is proved that the length of the quasi-period does not exceed degree of the fundamental $S$-unit of $L$. The problem of periodicity of continued fractions of elements of the form $\sqrt {f}/h^s$ for polynomials $f$ of even degree is more complicated. This is underlined by the example we found of a polynomial $f$ of degree $4$, for which the corresponding continued fractions have an abnormally large period length. Earlier in the article [5] we found examples of continued fractions of elements of the hyperelliptic field $L$ with a quasi-period length significantly exceeding the degree of the fundamental $S$-unit of $L$.