理想加尔丁多项式
Ideal Gårding polynomials
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中文总结 AI 辅助
研究理想加尔丁多项式，它是加尔丁多项式的凸性增强子类。通过建立通用模型、商凹性及相关不等式等方法，证明其具有强大结构理论，如在极化下不变等，还暗示了与凸几何和洛伦兹多项式的联系。
中文摘要 AI 辅助
我们引入了理想加尔丁多项式，它是加尔丁多项式的一个凸性增强子类，其加尔丁分量在偏导数下是递归凸的。该类严格包含实稳定多项式，经过平移和齐次化后属于洛伦兹类。我们的主要结果是，尽管有这种额外的凸性，理想加尔丁多项式仍具有强大的结构理论：它们在极化下保持不变，满足自然的闭包性质，并支持线性保持理论。本文的一个关键贡献是单变量加尔丁多项式的通用模型，由单调根序列描述，等价地由皮特曼 - 斯坦利多面体的体积多项式描述。我们建立了商凹性和牛顿 - 麦克劳林型不等式，这导致了极化定理，并暗示了与凸几何和洛伦兹多项式的进一步联系。
英文摘要
We introduce ideal Gårding polynomials, a convexity-enhanced subclass of Gårding polynomials whose Gårding components are recursively convex under partial differentiation. This class strictly contains real stable polynomials and, after translation and homogenization, lies in the Lorentzian class. Our main result is that ideal Gårding polynomials still admit a robust structure theory despite this additional convexity: they are preserved under polarization, satisfy natural closure properties, and support a linear preserver theory. A key contribution of this paper is a universal model for univariate Gårding polynomials, described by monotone root sequences and equivalently by volume polynomials of Pitman--Stanley polytopes. We establish quotient concavity, and Newton--Maclaurin type inequalities, which leads to the polarization theorem, and suggests further connections with convex geometry and Lorentzian polynomials.
