# Lorentzian polynomials

```@article{Branden2019LorentzianP,
title={Lorentzian polynomials},
author={Petter Brand'en and June Huh},
journal={arXiv: Combinatorics},
year={2019}
}```
• Published 2019
• Mathematics
• arXiv: Combinatorics
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 Hodge-Riemann relations for Lorentzian polynomials. Lorentzian polynomials are intimately connected to matroid theory and negative dependence properties… Expand
HODGE-RIEMANN RELATIONS AND LORENTZIAN POLYNOMIALS
• 2021
We introduce the theory of Lorentzian polynomials, motivated by the log-concavity properties that volume polynomials of projective varieties satisfy due to the validity of Hodge-Riemann relations. 1.Expand
Stability of combinatorial polynomials and its applications
• Mathematics
• 2021
Many important problems are closely related to the zeros of certain polynomials derived from combinatorial objects. The aim of this paper is to make a systematical study on the stability ofExpand
Positively hyperbolic varieties, tropicalization, and positroids
• Mathematics
• 2019
A variety of codimension \$c\$ in complex affine space is called positively hyperbolic if the imaginary part of any point in it does not lie in any positive linear subspace of dimension \$c\$. PositivelyExpand
Simplicial generation of Chow rings of matroids
• Mathematics
• 2019
We introduce a new presentation of the Chow ring of a matroid whose variables now admit a combinatorial interpretation via the theory of matroid quotients and display a geometric behavior analogousExpand
Lorentzian polynomials from polytope projections
• Mathematics
• Algebraic Combinatorics
• 2021
Lorentzian polynomials, recently introduced by Branden and Huh, generalize the notion of log-concavity of sequences to homogeneous polynomials whose supports are integer points of generalizedExpand
Lorentzian polynomials on cones and the Heron-Rota-Welsh conjecture
• Mathematics
• 2021
We give a short proof of the log-concavity of the coefficients of the reduced characteristic polynomial of a matroid. The proof uses an extension of the theory of Lorentzian polynomials to convexExpand
Lagrangian geometry of matroids
• Mathematics
• 2020
We introduce the conormal fan of a matroid M, which is a Lagrangian analog of the Bergman fan of M. We use the conormal fan to give a Lagrangian interpretation of the Chern-Schwartz-MacPherson cycleExpand
A G ] 2 8 A ug 2 02 0 CONIC STABILITY OF POLYNOMIALS AND POSITIVE MAPS
• 2020
Abstract. Given a proper cone K ⊆ Rn, a multivariate polynomial f ∈ C[z] = C[z1, . . . , zn] is calledK-stable if it does not have a root whose vector of the imaginary parts is contained in theExpand
Strict log-concavity of the Kirchhoff polynomial and its applications to the strong Lefschetz property
• Mathematics
• Journal of Algebra
• 2021
Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether theseExpand
Multivariate blowup-polynomials of graphs
• Mathematics
• 2021
In recent joint work (2021), we introduced a novel multivariate polynomial attached to every metric space – in particular, to every finite simple connected graph G – and showed it has severalExpand

#### References

SHOWING 1-10 OF 118 REFERENCES
Multivariate Pólya-Schur classification problems in the Weyl algebra
• Mathematics
• 2006
A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra A(n) thatExpand
Discrete Concavity and the Half-Plane Property
• P. Brändén
• Mathematics, Computer Science
• SIAM J. Discret. Math.
• 2010
A family of \$M-concave functions arising naturally from polynomials (over a field of generalized Puiseux series) with prescribed nonvanishing properties is introduced, which contains several of the most well studied \$M\$-conCave functions in the literature. Expand
Homogeneous multivariate polynomials with the half-plane property
• Mathematics, Computer Science
• 2004
It is proved that the support (set of nonzero coefficients) of a homogeneous multiaffine polynomial with the half-plane property is necessarily the set of bases of a matroid. Expand
Polynomials with the half-plane property and matroid theory
A polynomial f is said to have the half-plane property if there is an open half-plane H ⊂ C, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H . ThisExpand
Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions
• Mathematics
• 1984
1 Introduction to Locally Convex Topological Vector Spaces and Dual Pairs.- 1. Locally Convex Vector Spaces.- 2. Hahn-Banach Theorems.- 3. Dual Pairs.- Notes and Remarks.- 2 Radon Measures andExpand
Discrete convex analysis
• K. Murota
• Mathematics, Computer Science
• Math. Program.
• 1998
This work follows Rockafellar’s conjugate duality approach to convex/nonconvex programs in nonlinear optimization, while technically relying on the fundamental theorems of matroid-theoretic nature. Expand
Negative dependence and the geometry of polynomials
• Mathematics, Physics
• 2007
We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class coversExpand
Hodge Theory for Combinatorial Geometries
• Mathematics
• 2015
We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of Heron,Expand
Multivariate stable polynomials: theory and applications
Univariate polynomials with only real roots -- while special -- do occur often enough that their properties can lead to interesting conclusions in diverse areas. Due mainly to the recent work of twoExpand
Metric spaces and positive definite functions
As poo we get the space Em with the distance function maxi-, ... I xi X. Let, furthermore, lP stand for the space of real sequences with the series of pth powers of the absolute values convergent.Expand