Separating Cones defined by Toric Varieties: Some Properties and Open Problems

TL;DR

This paper investigates a hierarchy of cones between sums of squares and positive semidefinite forms, characterizing their properties, boundaries, and open problems, extending Hilbert's classical theorem using toric varieties.

Contribution

It introduces and analyzes a new cone filtration between sums of squares and positive forms, describing their closure, interior, boundary, and posing open problems related to dual cones and toric varieties.

Findings

Intermediate cones are closed.

Descriptions of interiors and boundaries of cones.

Open problems on dual cones and generalizations.

Abstract

In 1888, Hilbert proved that the cone $\mathcal{P}_{n+1,2d}$ of positive semidefinite forms in $n+1$ variables of degree $2d$ coincides with its subcone $\Sigma_{n+1,2d}$ of those forms that are representable as finite sums of squares if and only if $(n+1,2d) = (2,2d)_{d\geq1}$ or $(n+1,2)_{n\geq1}$ or $(3,4)$. These are the Hilbert cases. In [GHK23, GHK24], we applied the Gram matrix method to construct cones between $\Sigma_{n+1,2d}$ and $\mathcal{P}_{n+1,2d}$, defined by projective varieties containing the Veronese variety. In particular, we introduced and examined a specific cone filtration $$\Sigma_{n+1,2d} = C_0 \subseteq \ldots \subseteq C_n \subseteq C_{n+1} \subseteq \ldots \subseteq C_{k(n,d)-n} = \mathcal{P}_{n+1,2d}$$ and determined each strict inclusion in non-Hilbert cases. This gave us a refinement of Hilbert's 1888 theorem. Here, $k(n,d)+1$ is the dimension of the vector space of forms in $n+1$ variables of degree $d$. In this paper, we show that the intermediate cones $C_i$'s are closed and describe their interiors and boundaries. We discuss the membership problem for the $C_i$'s, present open problems concerning their dual cones and generalizations to cones defined by toric varieties.