Discriminant of system of equations

TL;DR

This paper defines a new discriminant for systems of algebraic equations, called the Euler discriminant, which generalizes classical discriminants and captures the topology of solution sets.

Contribution

It introduces the Euler discriminant as a new polynomial invariant for algebraic systems, linking topological, combinatorial, and algebraic properties.

Findings

The set of atypical systems forms a hypersurface in the space of all systems.

The Euler discriminant vanishes exactly at systems with singular solutions.

The Euler discriminant interpolates classical objects like resultants and discriminants.

Abstract

What polynomial in the coefficients of a system of algebraic equations should be called its discriminant? We prove a package of facts that provide a possible answer. Let us call a system typical, if the homeomorphic type of its set of solutions does not change as we perturb its (non-zero) coefficients. The set of all atypical systems turns out to be a hypersurface in the space of all systems of k equations in n variables, whose monomials are contained in k given finite sets. This hypersurface B contains all systems that have a singular solution, this stratum is conventionally called the discriminant, and the codimension of its components has not been fully understood yet (e.g. dual defect toric varieties are not classified), so the purity of dimension of B looks somewhat surprising. We deduce it from a similar tropical purity fact. A generic system of equations in a component B_i of the hypersurface B differs from a typical system by the Euler characteristic of its set of solutions. Regarding the difference of these Euler characteristics as the multiplicity of B_i, we turn B into an effective divisor, whose equation we call the Euler discriminant by the following reasons. Firstly, it vanishes exactly at those systems that have a singular solution (possibly at infinity). Secondly, despite its topological definition, there is a simple linear-algebraic formula for it, and a positive formula for its Newton polytope. Thirdly, it interpolates many classical objects (sparse resultant, A-determinant, discriminant of deformation) and inherits many of their nice properties. This allows to specialize our results to generic polynomial maps: the bifurcation set of a dominant polynomial map, whose components are generic linear combinations of finitely many monomials, is always a hypersurface, and a generic atypical fiber of such a map differs from a typical one by its Euler characteristic.