On the Period Length of a Functional Continued Fraction over a Number Field

Abstract

In the classical case, the connection between the periodicity of the continued fraction of $$\sqrt f $$ and the existence of a fundamental unit of the corresponding hyperelliptic field $$\mathcal{L} = K(x)(\sqrt f )$$ , where K is a field of characteristic different from 2, has long been known. For the element $$\sqrt f $$ , the period length of the continued fraction in $$K((1{\text{/}}x))$$ can be trivially estimated from above by the doubled degree of the fundamental unit. Much more complicated and interesting is the problem of estimating (from above) the period length of other elements of $$\mathcal{L}$$ that have a periodic continued fraction. Among these elements, those of the form $$\sqrt f {\text{/}}{{x}^{s}}$$ , $$s \in \mathbb{Z}$$ , play a key role. For such elements, the period length can be many times greater than the double degree of the fundamental unit. In this article, we find upper bounds for the period length of key elements of hyperelliptic fields $$\mathcal{L}$$ over number fields K. An example is found that demonstrates the sharpness of the proven upper bounds.