Components of discriminants for systems of equations and irreducibility of determinants

Source abstract

Abstract The discriminant of a multivariate polynomial with indeterminate coefficients is not necessarily a hypersurface, and the problem of characterizing its codimension remained open for an extended period. We solve this problem for the discriminants of systems of polynomials with indeterminate coefficients in which the number of equations equals the number of unknowns (square polynomial systems). Such discriminants 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 prove Esterov’s conjecture that the discriminant of a nonlinear irreducible square polynomial system is an irreducible hypersurface. Building on this result, we enumerate all components and determine their dimensions and degrees for each of the three conventional formalizations of the notion of a discriminant in this setting (mixed, Cayley, and $A$-discriminants), for square systems and for those overdetermined systems that do not reduce to underdetermined ones. Both the proof of the Esterov conjecture and the descriptions of the three types of discriminants are based on the theory of polymatroids.