Lorentzian polynomials on cones
Abstract
Inspired by the theory of hyperbolic polynomials and Hodge theory, we develop the theory of Lorentzian polynomials on cones. This notion captures the HodgeRiemann 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 HeronRotaWelsh conjecture for the characteristic polynomial of a matroid, and the AlexandrovFenchel inequalities for convex bodies. We then characterize Chow rings of simplicial fans which satisfy the HodgeRiemann 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2023
 DOI:
 10.48550/arXiv.2304.13203
 arXiv:
 arXiv:2304.13203
 Bibcode:
 2023arXiv230413203B
 Keywords:

 Mathematics  Combinatorics