Obstructions to determinantal representability

TL;DR

This paper disproves a conjecture that all real zero polynomials have determinantal representations with symmetric matrices, by linking LMI representability to polymatroid representability and using recent matroid theory results.

Contribution

It shows that not all real zero polynomials admit determinantal representations, challenging the conjecture and connecting LMI representability to polymatroid theory.

Findings

Disproved Helton and Vinnikov's conjecture.

Identified a real zero polynomial without a determinantal power.

Linked LMI representability to polymatroid representability.

Abstract

There has recently been ample interest in the question of which sets can be represented by linear matrix inequalities (LMIs). A necessary condition is that the set is rigidly convex, and it has been conjectured that rigid convexity is also sufficient. To this end Helton and Vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices. We disprove this conjecture. By relating the question of finding LMI representations to the problem of determining whether a polymatroid is representable over the complex numbers, we find a real zero polynomial such that no power of it admits a determinantal representation. The proof uses recent results of Wagner and Wei on matroids with the half-plane property, and the polymatroids associated to hyperbolic polynomials introduced by Gurvits.