On types of degenerate critical points of real polynomial functions

TL;DR

This paper introduces a method to classify degenerate critical points of multivariate polynomial functions by defining a faithful radius and analyzing the function's extrema within that radius.

Contribution

The paper presents a novel concept of faithful radius and algorithms to determine the type of degenerate critical points of polynomial functions.

Findings

Faithful radius can be used to classify critical points.

Algorithms effectively determine the type of degenerate critical points.

Method applies to multivariate polynomial functions.

Abstract

In this paper, we consider the problem of identifying the type (local minimizer, maximizer or saddle point) of a given isolated real critical point $c$, which is degenerate, of a multivariate polynomial function $f$. To this end, we introduce the definition of faithful radius of $c$ by means of the curve of tangency of $f$. We show that the type of $c$ can be determined by the global extrema of $f$ over the Euclidean ball centered at $c$ with a faithful radius.We propose algorithms to compute a faithful radius of $c$ and determine its type.