Collision Positivity for Three-Variable Symmetric Monomial Inequalities

Abstract

Let \(λ\succγ\succμ\) be equal-degree exponent partitions with at most three parts, and let \begin{equation*} P_{λ,γ,μ} = J_λ+J_μ-2J_γ, \end{equation*} where \(J_ν\) denotes the symmetric monomial orbit sum associated with \(ν\). We prove a necessary and sufficient collision criterion for the positivity of this three-point majorization difference in three variables. For nonnegative integer exponent partitions, \begin{equation*} P_{λ,γ,μ}(x,y,z)\ge0 \qquad(x,y,z>0) \end{equation*} if and only if \begin{equation*} P_{λ,γ,μ}(t,1,1)\ge0 \qquad(t>0). \end{equation*} Thus the positivity of a genuinely three-variable symmetric polynomial of this form is completely determined by its one-variable restriction to the locus where two variables coincide. The theorem gives a uniform and effectively checkable criterion for an infinite class of symmetric polynomial inequalities. For a fixed integer chain, the global three-variable problem is reduced to a single univariate polynomial inequality, which can often be verified exactly by factorization, Sturm's theorem, or other one-variable methods. The criterion extends well beyond the classical Schur family, produces explicit inequalities outside the Schur pattern, and yields genuine strengthenings and refinements of Schur's inequality.