Hyperelliptic continued fractions in the singular case of genus zero

Given a polynomial of even degree D(t) with complex coefficients, we consider the continued fraction expansion of root 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 root 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 showthat there are non-trivial polynomials D(t) with non-periodic expansions such that infinitely many partial quotients have degree greater than one.