Research

Abeer Al Ahmadieh

submitted

This paper explores the fibers of the principal minor map over a general field. The principal minor map is the map that assigns to each n×n matrix the 2ⁿ-vector of its principal minors. In 1984, Hartfiel and Loewy proposed a condition that was sufficient to ensure that the fiber of the principal minor map is a single point up to diagonal equivalence. Loewy later improved upon this condition in 1986. In this paper, we provide a necessary and sufficient condition for the fiber to be a point up to diagonal equivalence. Additionally, we establish a connection between the reducibility of a matrix and the reducibility of its determinantal representation. Using this connection, we fully characterize the fiber of symmetric and Hermitian matrices in the space of n×n matrices over any field F. We also use these techniques to answer a question of Borcea, Brändén, and Liggett concerning real stable matrices.

Abeer Al Ahmadieh and Cynthia Vinzant

submitted

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 (SL₂(R))ⁿxS_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 F, there is no finite set of equations whose orbit under (SL₂(F))ⁿxS_n cuts out the image of n×n matrices over F under the principal minor map for every n.