Non-negative polynomials on generalized elliptic curves
Functional Analysis
2026-01-28 v3 Algebraic Geometry
Classical Analysis and ODEs
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.
Cite
@article{arxiv.2508.13850,
title = {Non-negative polynomials on generalized elliptic curves},
author = {Mario Kummer and Aljaž Zalar},
journal= {arXiv preprint arXiv:2508.13850},
year = {2026}
}
Comments
11 pages; in this version results of v1 are extended to generalized elliptic curves and presented in the projective setting; furthermore, reference issues from v2 are fixed in this version