Periodicity Criterion for Continued Fractions of Key Elements in Hyperelliptic Fields

Abstract

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 numerical continued fractions of elements of quadratic fields. It is known that the periodicity of a continued fraction of $\sqrt f {\text{/}}{{h}^{{g + 1}}}$ constructed using the valuation associated with a first-degree polynomial h is equivalent to the existence of nontrivial S-units in a field L of genus g and is equivalent to the existence of nontrivial torsion in the divisor class group. In this article, we find an exact interval of values of $s \in \mathbb{Z}$ such that the elements $\sqrt f {\text{/}}{{h}^{s}}$ have a periodic continued fraction expansion, where $f \in \mathbb{Q}[x]$ is a square-free polynomial of even degree. For polynomials f of odd degree, the periodicity problem for continued fractions of elements of the form $\sqrt f {\text{/}}{{h}^{s}}$ was discussed in [5], where it was proved that the length of the quasi-period does not exceed the degree of the fundamental S-unit of L. For polynomials f of even degree, the periodicity of continued fractions of elements of the form $\sqrt f {\text{/}}{{h}^{s}}$ is a more complicated problem. This is underlined by an example we have found, namely, a polynomial f of degree 4 for which the corresponding continued fraction has an abnormally long period. Earlier in [5], for elements of a hyperelliptic field L, we found examples of continued fractions with a quasi-period length significantly exceeding the degree of the fundamental S-unit of L.