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.

We prove that every lower unitriangular and totally nonnegative matrix gives rise to a family of polynomials with only real zeros. This has consequences for problems in several areas of mathematics. We use it to develop a general theory for chain enumeration in posets and zeros of chain polynomials. The results obtained extend and unify results of the first author, Brenti, Welker and Athanasiadis. In the process we define a notion of h -vector for a large class of posets which generalize the notions of h -vectors associated to simplicial and cubical complexes. A consequence of our methods is a characterization of the convex hull of all characteristic polynomials of hyperplane arrangements of fixed dimension and over a fixed finite field. This may be viewed as a refinement of the Critical Problem of Crapo and Rota. We also use the methods developed to solve an open problem posed by Forgács and Tran on the real-rootedness of polynomials arising from certain bivariate rational functions.

Given a linear ordinary differential operator T with polynomial coefficients, we study the class of closed subsets of the complex plane such that T sends any polynomial (respectively, any polynomial of degree exceeding a given positive integer) with all roots in a given subset to a polynomial with all roots in the same subset or to 0. Below we discuss some general properties of such invariant subsets, as well as the problem of existence of the minimal under inclusion invariant subset.

We prove that the chain polynomial of the lattice of flats of a paving matroid is real-rooted, and we define a class of matroids called generalized paving matroids. Generalized paving matroids associated to subspace lattices are shown to have real-rooted chain polynomials, by a study of a q-analog of the subdivision operator. We finish by studying single element extensions, and prove that the chain polynomials of the lattice of flats of single element extensions of (Formula presented.) and (Formula presented.) are real-rooted.

We present a new lower bound on the number of contingency tables, improving upon and extending previous lower bounds by Barvinok [Bar09, Bar16] and Gurvits [Gur15]. As an application, we obtain new lower bounds on the volumes of flow and transportation polytopes. Our proofs are based on recent results on Lorentzian polynomials. 

Eulerian polynomials are fundamental in combinatorics and algebra. In this paper we study the linear transformation A : R[t] -> R[t] defined by A(t(n)) = A(n)(t), where A(n)(t) denotes the n-th Eulerian polynomial. We give combinatorial, topological and Ehrhart theoretic interpretations of the operator A, and investigate questions of unimodality and real-rootedness. In particular, we disprove a conjecture by Brenti (1989) concerning the preservation of real zeros, and generalize and strengthen recent results of Haglund and Zhang (2019) on binomial Eulerian polynomials.

We prove that projective spaces of Lorentzian and real stable polynomials are homeomorphic to Euclidean balls. This solves a conjecture of June Huh and the author. The proof utilises and refines a connection between the symmetric exclusion process in interacting particle systems and the geometry of polynomials.

In algebraic, topological, and geometric combinatorics, inequalities among the coefficients of combinatorial polynomials are frequently studied. Recently, a notion called the alternatingly increasing property, which is stronger than unimodality, was introduced. In this paper, we relate the alternatingly increasing property to real-rootedness of the symmetric decomposition of a polynomial to develop a systematic approach for proving the alternatingly increasing property for several classes of polynomials. We apply our results to strengthen and generalize real-rootedness, unimodality, and alternatingly increasing results pertaining to colored Eulerian and derangement polynomials, Ehrhart h*-polynomials for lattice zonotopes, h-polynomials of barycentric subdivisions of doubly Cohen-Macaulay level simplicial complexes, and certain local h-polynomials for subdivisions of simplices. In particular, we prove two conjectures of Athanasiadis.

We introduce and study s-lecture hall P-partitions which is a generalization of s-lecture hall partitions to labeled (weighted) posets. We provide generating function identities for s-lecture hall P-partitions that generalize identities obtained by Savage and Schuster for s-lecture hall partitions, and by Stanley for P-partitions. We also prove that the corresponding (P, s)-Eulerian polynomials are real-rooted for certain pairs (P, s), and speculate on unimodality properties of these polynomials.

We study the class of Lorentzian polynomials. The class contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. We prove that the Hessian of a nonzero Lorentzian polynomial has exactly one positive eigenvalue at any point on the positive orthant. This property can be seen as an analog of the Hodge-Riemann relations for Lorentzian polynomials. Lorentzian polynomials are intimately connected to matroid theory and negative dependence properties. We show that matroids, and more generally M-convex sets, are characterized by the Lorentzian property, and develop a theory around Lorentzian polynomials. In particular, we provide a large class of linear operators that preserve the Lorentzian property and prove that Lorentzian measures enjoy several negative dependence properties. We also prove that the class of tropicalized Lorentzian polynomials coincides with the class of M-convex functions in the sense of discrete convex analysis. The tropical connection is used to produce Lorentzian polynomials from M-convex functions. We give two applications of the general theory. First, we prove that the homogenized multivariate Tutte polynomial of a matroid is Lorentzian whenever the parameter q satisfies 0 < q <= 1. Consequences are proofs of the strongest Mason's conjecture from 1972 and negative dependence properties of the random cluster model in statistical physics. Second, we prove that the multivariate characteristic polynomial of an M-matrix is Lorentzian. This refines a result of Holtz who proved that the coefficients of the characteristic polynomial of an M-matrix form an ultra log-concave sequence.