2023/04/26 by Petter Brändén, Brändén, Petter, Jonathan R. Leake +1 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2304.13203
openalex publication_date 2023/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Inspired by the theory of hyperbolic polynomials and Hodge theory, we develop the theory of Lorentzian polynomials on cones. This notion captures the Hodge-Riemann relations of degree zero and one. Motivated by fundamental properties of volume polynomials of Chow rings of simplicial fans, we define a class of multivariate polynomials which we call hereditary polynomials. We give a complete and easily checkable characterization of hereditary Lorentzian polynomials. This characterization is used to give elementary and simple proofs of the Heron-Rota-Welsh conjecture for the characteristic polynomial of a matroid, and the Alexandrov-Fenchel inequalities for convex bodies. We then characterize Chow rings of simplicial fans which satisfy the Hodge-Riemann relations of degree zero and one, and we prove that this property only depends on the support of the fan. Several different characterizations of Lorentzian polynomials on cones are provided.