On the fibers of the principal minor map and an application to stable polynomials

Abstract

This paper explores the fibers of the principal minor map over an arbitrary field. The principal minor map assigns to each $n\times n$ matrix the $2^n$-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 fibers that contain a symmetric or Hermitian matrix in the space of $n\times n$ matrices over any field. We also use these techniques to answer a question of Borcea, Brändén, and Liggett concerning real stable matrices.