Real Inflection Points of Real Linear Series on an Elliptic Curve

Abstract

Given a real elliptic curve E with non-empty real part and a real effective divisor $\mathcal{D}$ on E arising via pullback from ${\mathbb{P}}^1$ under the hyperelliptic structure map, we study the real inflection points of distinguished subseries of the complete real linear series $\left|\mathcal{D}\right|$ on E. We define inflection 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 hyperelliptic curves to a metrized complex of curves with elliptic curve components that we carried out in an earlier joint work with I. Biswas.

Keywords:

• linear series
• real algebraic curves
• Wronskians

Acknowledgments

The authors are grateful to Eduardo Ruiz-Duarte for his valuable help with the computations, and to the anonymous referee, whose suggestions have helped us improve the exposition.

Declaration of interest

No potential conflict of interest was reported by the authors.