Diagonalizations of denormalized volume polynomials

TL;DR

This paper proves that diagonalization preserves the class of denormalized volume polynomials, and more generally that products, normalization, and lower truncation preserve this property. The authors develop a framework showing diagonalization of a denormalized volume polynomial remains denormalized, with corollaries extending to products and truncations and connections to Lorentzian polynomials and the reverse Khovanskii-Teissier inequality. They provide a geometric interpretation and show how these results yield new inequalities for intersection numbers. As an application, they demonstrate that the Kahn-Saks polynomial associated with a poset is a denormalized volume polynomial, leading to concrete inequalities for chain-interval statistics and linking to Rayleigh-type properties of the normalized polynomial.

Abstract

We show that diagonalization, products and lower truncations preserve the property of being a denormalized volume polynomial. We also discuss an application to poset inequalities.

Theorems & Definitions (50)

• Theorem 1.1: Diagonalization
• Theorem 1.2: Products, =Corollary \ref{['cor:denorm-product']}
• Theorem 1.3: Normalization, $\subset$ Corollary \ref{['cor:normalizationpreservesvolume']}
• Theorem 1.4: Truncation, =Corollary \ref{['cor:truncation']}
• Remark 1.5
• Definition 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Lemma 2.4