Non-negative polynomials on generalized elliptic curves

TL;DR

This paper investigates the structure of non-negative polynomials on generalized elliptic curves, revealing properties of their extreme rays and convex hulls, and generalizes existing results on 2-torsion points to broader classes of curves.

Contribution

It introduces new insights into the cone of non-negative polynomials on generalized elliptic curves and extends a key result on 2-torsion points to singular and reducible curves.

Findings

Zero set of every extreme ray has dense real points

Convex hull of real points is a spectrahedron when embedded via a complete linear system

Generalizes Geyer--Martens result to singular and reducible curves

Abstract

We study the cone of non-negative polynomials on generalized elliptic curves. We show that the zero set of every extreme ray has dense real points. If a generalized elliptic curve is embedded via a complete linear system, then we show that the convex hull of its real points (taken inside any affine chart containing all real points) is a spectrahedron. On the way, we generalize a result by Geyer--Martens on 2-torsion points in the Picard group of smooth real curves (of arbitrary genus) to possibly singular and reducible ones.