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

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