Components of discriminants for systems of equations and irreducibility of determinants

TL;DR

This paper characterizes the components and irreducibility of discriminants for square polynomial systems, resolving longstanding open problems and conjectures using polymatroid theory.

Contribution

It provides a complete characterization of discriminant components and proves Esterov's conjecture for square polynomial systems, linking discriminant irreducibility to matrix subspace properties.

Findings

Discriminant components are fully classified for square polynomial systems.

Esterov's conjecture on irreducibility of discriminants is proven.

Explicit descriptions of discriminants for various formalizations are provided.

Abstract

The discriminant of a multivariate polynomial with indeterminate coefficients is not necessarily a hypersurface, and characterizing its codimension was an open problem for quite a while. We resolve this problem for the discriminants of systems of polynomials with indeterminate coefficients and with the same number of equations and unknowns (square polynomial systems). This version is more involved in the sense that the discriminant may have several components of different dimensions. In the space of square matrices, we characterize row-generated subspaces on which the determinant is an irreducible polynomial. This allows us to resolve the Esterov conjecture for square polynomial systems whose discriminant is an irreducible hypersurface. Based on this result, we enumerate all the components and determine their dimensions and degrees for each of the three conventional ways to formalize the notion of a discriminant in this setting (mixed, Cayley, and A-discriminants) in cases of square and overdetermined systems. The proof of Esterov's conjecture and descriptions of the three types of discriminants are based on the theory of polymatroids.