Non-negative polynomials on generalized elliptic curves
Mario Kummer, Aljaž Zalar
TL;DR
This work analyzes the cone $P(X,L)$ of non-negative sections on generalized elliptic curves $X$ over $\mathbb{R}$, showing that extreme rays have real zeros dense and that, when $L$ embeds $X$ via a complete linear system, the convex hull of $X(\mathbb{R})$ is a spectrahedron. It provides a sums-of-squares framework: every non-negative section is expressible as a sum of squares of sections of a coherent sheaf $\mathcal{G}$, and the dual cone $P(X,L)^{\vee}$ is described by a positive semidefinite bilinear form $B_\ell$ on these sections, yielding a block-structured spectrahedral description. The paper extends Geyer--Martens’ $2$-torsion analysis to singular (and reducible) real curves, offering divisibility and torsion-count results, and links these to geometric properties of convex hulls and extreme rays. Overall, it bridges real algebraic geometry with convex semidefinite programming, providing concrete structural insights and computable descriptions for a broad class of real curves, including generalized elliptic curves and their singular degenerations.
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.