Hyperelliptic continued fractions in the singular case of genus zero

TL;DR

This paper explores the continued fraction expansion of square roots of polynomials in genus zero curves, establishing explicit geometric conditions for partial quotient degrees and revealing non-periodic expansions with infinitely many high-degree partial quotients.

Contribution

It provides a detailed analysis of hyperelliptic continued fractions in genus zero cases, linking partial quotient degrees to geometric conditions and demonstrating non-trivial non-periodic expansions.

Findings

Explicit geometric conditions for partial quotient degrees

Existence of non-trivial polynomials with non-periodic expansions

Infinitely many partial quotients with degree greater than one

Abstract

It is possible to define a continued fraction expansion of elements in a function field of a curve by expanding as a Laurent series in a local parameter. Considering the square root of a polynomial $\sqrt{D(t)}$ leads to an interesting theory related to polynomial Pell equations. Unlike the classical Pell equation, the corresponding polynomial equation is not always solvable and its solvability is related to arithmetic conditions on the Jacobian (or generalized Jacobian) of the curve defined by $y^2=D(t)$. In this setting, it has been shown by Zannier in \cite{zannier} that the sequence of the degrees of the partial quotients of the continued fraction expansion of $\sqrt{D(t)}$ is always 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.