The 'Core' of Symmetric Homogeneous Polynomial Inequalities of Degree Four of Three Real Variables

TL;DR

This paper investigates symmetric homogeneous polynomial inequalities of degree four in three real variables, identifying cases where minimal coefficients are independent, using a parametric approach to resolve the problem.

Contribution

It introduces a parametric representation to analyze when the smallest coefficients in these inequalities are not determined by others.

Findings

Characterization of cases with independent minimal coefficients

Development of a parametric method for inequality analysis

Resolution of the inequality problem using the new approach

Abstract

In this paper we explore inequalities between symmetric homogeneous polynomials of degree four of three real variables and three nonnegative real variables. The main theorems describe the cases in which the smallest possible coefficient is not expressed by the other coefficients. The problem is resolved by introducing a parametric representation.