Multi-valued hyperelliptic continued fractions of generalized Halphen type

TL;DR

This paper extends the theory of continued fractions to higher genus hyperelliptic cases, exploring their properties, symmetries, and periodicity, thus broadening the understanding of generalized Halphen continued fractions.

Contribution

It introduces higher genus hyperelliptic continued fractions, analyzing their structure, symmetry, and periodicity, which is a novel generalization of classical Halphen theory.

Findings

Hyperelliptic continued fractions exhibit symmetry properties.

Periodic structures are identified in higher genus cases.

New classifications of regular and irregular hyperelliptic elements.

Abstract

We introduce and study higher genera generalizations of the Halphen theory of continued fractions. The basic notion we start with is hyperelliptic Haplhen (HH) element $$\frac{\sqrt{X_{2g+2}}-\sqrt{Y_{2g+2}}}{x-y},$$ depending on parameter $y$, where $X_{2g+2}$ is a polynomial of degree $2g+2$ and $Y_{2g+2}=X_{2g+2}(y)$. We study regular and irregular HH elements. their continued fraction development and some basic properties of such development: even and odd symmetry and periodicity.