Determinantal representations and the image of the principal minor map

TL;DR

This paper characterizes when multiaffine polynomials have Hermitian determinantal representations and describes the algebraic structure of the principal minor map's image for Hermitian matrices, using factorization of Rayleigh differences.

Contribution

It provides a new characterization of Hermitian determinantal representations via Rayleigh differences and describes the algebraic structure of the principal minor map's image for Hermitian matrices.

Findings

Hermitian determinantal representations characterized by Rayleigh difference factorizations

The image of Hermitian matrices under the principal minor map is described by finitely many equations and inequalities

No finite set of equations can describe the principal minor map image over arbitrary fields for all matrix sizes

Abstract

In this paper we explore determinantal representations of multiaffine polynomials and consequences for the image of various spaces of matrices under the principal minor map. We show that a real multiaffine polynomial has a definite Hermitian determinantal representation if and only if all of its so-called Rayleigh differences factor as Hermitian squares and use this characterization to conclude that the image of the space of Hermitian matrices under the principal minor map is cut out by the orbit of finitely many equations and inequalities under the action of $({\rm SL}_2(\mathbb{R}))^{n} \rtimes S_{n}$. We also study such representations over more general fields with quadratic extensions. Factorizations of Rayleigh differences prove an effective tool for capturing subtle behavior of the principal minor map. In contrast to the Hermitian case, we give examples to show for any field $\mathbb{F}$, there is no finite set of equations whose orbit under $({\rm SL}_2(\mathbb{F}))^{n} \rtimes S_{n}$ cuts out the image of $n\times n$ matrices over $\mathbb{F}$ under the principal minor map for every $n$.