三变量对称单项式不等式的碰撞正性
Collision Positivity for Three-Variable Symmetric Monomial Inequalities
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中文总结 AI 辅助
本文证明三变量对称单项式不等式正性等价于其两变量相等处的单变量限制非负，从而将无穷类不等式约化为单变量检验，并推广了 Schur 不等式。
中文摘要 AI 辅助
设 \\(\lambda\succ\gamma\succ\mu\\) 为至多三部分且次数相等的指数分拆，并设 \\(P_{\lambda,\gamma,\mu} = J_\lambda+J_\mu-2J_\gamma\\)，其中 \\(J_\nu\\) 表示与 \\(\nu\\) 关联的对称单项式轨道和。我们证明了三变量中该三点优序差正性的一个充要碰撞判据。对于非负整数指数分拆，\\(P_{\lambda,\gamma,\mu}(x,y,z)\ge0\\)（对所有 \\(x,y,z>0\\)）当且仅当 \\(P_{\lambda,\gamma,\mu}(t,1,1)\ge0\\)（对所有 \\(t>0\\)）。因此，这种形式的真正三变量对称多项式的正性完全由其限制在两个变量相等处的单变量限制所决定。该定理为无穷类对称多项式不等式给出了一个统一且可有效检验的判据。对于固定的整数链，全局三变量问题被约化为单个单变量多项式不等式，后者通常可通过因式分解、Sturm 定理或其他单变量方法精确验证。该判据远远超出了经典 Schur 族，在 Schur 模式之外产生了显式不等式，并给出了 Schur 不等式的真正加强和细化。
英文摘要
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.
