S-units and periodicity of square root in hyperelliptic fields

Abstract

For a polynomial f of odd degree, nontrivial S-units can be effectively related to the continued fraction expansion of elements involving the square root of the polynomial f only in the case where S consists of an infinite valuation and a finite valuation determined by a first-degree polynomial h. In the paper, the proof that the quasi-periodicity of the continued fraction expansion of an element of the form \(\frac{{\sqrt f }}{{{h^s}}}\) implies periodicity is completed. In particular, it is proved that the continued fraction expansion of \(\sqrt f \) for f of any degree is quasi-periodic in k((h)) it and only if it is periodic.