The criterion of periodicity of 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 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.