Kolloquien und Oberseminare Übersicht Detail

Vortragende Person/Vortragende Personen:
Sarah Hess

For $n,d\in\mathbb{N}$, let $\mathcal{P}_{n+1,2d}$ denote the convex cone of positive semi-definite (PSD) homogeneous polynomials (forms) in $n+1$ variables of degree $2d$ with real coefficients. This cone contains the convex cone $\Sigma_{n+1,2d}$ of forms in $n+1$ variables of degree $2d$ with real coefficients that are representable as finite sums of squares (SOS) of forms of half degree $d$. By Hilbert's 1888 Theorem, $\Sigma_{n+1,2d}=\mathcal{P}_{n+1,2d}$ exactly in the Hilbert cases $(n+1,2d)$ with $n+1=2$ or $2d=2$ or $(3,4)$.
        
In this talk, we introduce, for the non-Hilbert cases, a specific cone filtration \begin{equation} \label{abs:eq1} \Sigma_{n+1,2d}=C_0 \subseteq C_1 \subseteq \ldots \subseteq C_{k(n,d)-n-1} \subseteq C_{k(n,d)-n}=\mathcal{P}_{n+1,2d}, \end{equation} (1)
defined via the Gram matrix method by a filtration of $k(n,d)-n+1$ irreducible projective varieties containing the Veronese variety, where $k(n,d):=\dim(\mathcal{F}_{n+1,d})-1$. By examining this cone filtration, we determine the number $\mu(n,d)$ of strictly separating intermediate cones (i.e., $C_i$ such that $\Sigma_{n+1,2d}\subsetneq C_i \subsetneq \mathcal{P}_{n+1,2d}$) in (1) and, thus, reduce (1) to a specific cone subfiltration \begin{equation} \label{abs:eq2} \Sigma_{n+1,2d}=C_0^\prime\subsetneq C_1^\prime \subsetneq \ldots \subsetneq C_{\mu(n,d)}^\prime \subsetneq C_{\mu(n,d)+1}^\prime=\mathcal{P}_{n+1,2d}, \end{equation} (2)
in which each inclusion is strict. Note that each $C_i^\prime$ in (2) is a convex semialgebraic set.

In 2009, Helton and Nie conjectured that any convex semialgebraic set (over $\mathbb{R}$) is a spectrahedral shadow (i.e., the image of a spectrahedron under an affine-linear map). For instance, $\Sigma_{n+1,2d}$ is a spectrahedral shadow and for many years no counterexample of a convex semialgebraic set that is not a spectrahedral shadow was known. In 2018, Scheiderer succeded to develop a method to produce convex semialgebraic sets that are not spectrahedral shadows. In particular, he showed that $\mathcal{P}_{n+1,2d}$ is not a spectrahedral shadow in non-Hilbert cases.
    
In this talk, for $i=1,\ldots,\mu(n,d)$, we show that each $C_i^\prime$ in (2), and hence each strictly separating $C_i$ in (1), fails to be a spectrahedral shadow. To this end, we construct explicit examples of separating forms $f\in C^\prime_i\backslash C^\prime_{i-1}$. This allows us to apply the methods of Scheiderer to prove that, for $i=1,\ldots,\mu(2,3)$, $C_i^\prime$ is not a spectrahedral shadow as a subcone of $\mathcal{P}_{3,6}$. Similarly, we also show that, for $i=1,\ldots,\mu(n,2)$, $C_i^\prime$ is not a spectrahedral shadow as a subcone of $\mathcal{P}_{n+1,4}$. Lastly, we develop an inductive argument (over $d$) that allows us to verify this assertion for $n\geq 2$, $d\geq 3$.
This is a joint work with Charu Goel and Salma Kuhlmann.