Hyperelliptic continued fractions in the singular case of genus zero

Abstract

Given a polynomial of even degree $D(t)$ with complex coefficients, we consider the continued fraction expansion of $\sqrt{D(t)}$. In this setting, it has been shown by Zannier that the sequence of the degrees of the partial quotients of the continued fraction expansion of $\sqrt{D(t)}$ is eventually periodic, even when the expansion itself is not. In this article, we work out in detail the case in which the curve $y^2=D(t)$ has genus $0$, establishing explicit geometric conditions corresponding to the appearance of partial quotients of certain degrees in the continued fraction expansion. We also show that there are non-trivial polynomials $D(t)$ with non-periodic expansions such that infinitely many partial quotients have degree greater than one.