Real inflection points of real linear series on an elliptic curve

TL;DR

This paper investigates the real inflection points of certain linear series on real elliptic curves, introducing key polynomials that encode inflection points and relate to the curve's ramification structure.

Contribution

It defines key polynomials to index inflection points on real elliptic curves and establishes a recursive hierarchy similar to division polynomials, extending previous degeneration analyses.

Findings

Key polynomials index inflection points away from ramification points.

Recursive hierarchy of polynomials parallels division polynomials.

Enhanced understanding of inflection loci in real elliptic curves.

Abstract

Given a real elliptic curve $E$ with non-empty real part and $[D]\in \mbox{Pic}^2 E$ its $g_2^1$, we study the real inflection points of distinguished subseries of the complete real linear series $|\mathcal{L}_\mathbb{R}(kD)|$ for $k\geq 3$. We define {\it key polynomials} whose roots index the ($x$-coordinates of) inflection points of the linear series, away from the points where $E$ ramifies over $\mathbb{P}^1$. These fit into a recursive hierarchy, in the same way that division polynomials index torsion points. Our study is motivated by, and complements, an analysis of how inflectionary loci vary in the degeneration of real {\it hyperelliptic} curves to a metrized complex of curves with elliptic curve components that we carried out in our previous article with Biswas.