Algebraicity of the division points of the trifolium and related topics

TL;DR

This paper generalizes classical results on algebraic division points from circles and lemniscates to the Erdős lemniscate with three leaves, exploring algebraicity and transcendence properties via hyperelliptic curve analysis.

Contribution

It extends algebraic division point results to the three-leaf Erdős lemniscate and investigates related algebraicity and transcendence questions using hyperelliptic Jacobian structures.

Findings

Division points on the Erdős lemniscate can be algebraic under certain conditions.

The length of specific polynomial lemniscates can be transcendental.

Analysis of hyperelliptic Jacobians reveals structural insights into these properties.

Abstract

Gauss and Abel proved that the points dividing the unit circle and the lemniscate of Bernoulli in parts of equal length have algebraic coordinates. In this note we generalise these results to the Erd\H{o}s lemniscate with three leaves. We also study further questions related to the algebraicity of division points and transcendence of length of a class of curves including polynomial lemniscates. To do this we analyse the structure and periods of the Jacobian of certain hyperelliptic curves.