Irreducibility of determinants, and Esterov's conjecture on $\mathscr{A}$-discriminants

TL;DR

This paper characterizes when determinants are irreducible in matrix spaces and resolves Esterov's conjecture on the irreducibility of discriminants of square polynomial systems, advancing understanding of algebraic properties of these discriminants.

Contribution

It provides a complete characterization of subspaces with irreducible determinants and proves Esterov's conjecture on the irreducibility of discriminants for square polynomial systems.

Findings

Characterization of row-generated subspaces with irreducible determinants

Proof of Esterov's conjecture on irreducibility of discriminants

Implications for the structure of polynomial system discriminants

Abstract

In the space of square matrices, we characterize row-generated subspaces, on which the determinant is an irreducible polynomial. As a corollary, we characterize square systems of polynomial equations with indeterminate coefficients, whose discriminant is an irreducible hypersurface. This resolves a conjecture of Esterov, and, in a sequel paper, leads to a complete description of components and codimensions for discriminants of square systems of equations.