Multiple discriminants and critical values of a multivariate polynomial

TL;DR

This paper derives an equation for the critical values of multivariate polynomials, providing a new method to analyze their critical points and values in a systematic way.

Contribution

It introduces a novel equation for critical values of multivariate polynomials, expanding understanding of polynomial critical points and their values.

Findings

Derived an explicit equation for polynomial critical values

Enhanced analysis of multivariate polynomial critical points

Provides a foundation for further algebraic and geometric studies

Abstract

A critical value of a function is the value of this function at one of its critical points. Each critical point of a differentiable multivariate function is described by the equations which consist in equating to zero all of its partial derivatives. However, in general case there is no equation for the corresponding critical value. The case of polynomials is different. In the present paper an equation for critical values of a polynomial is derived.