# Lorentzian polynomials

### Abstract

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 $\mathrm {M}$-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 $\mathrm {M}$-convex functions in the sense of discrete convex analysis. The tropical connection is used to produce Lorentzian polynomials from $\mathrm {M}$-convex functions.

We give two applications of the general theory. First, we prove that\nonbreakingspace the homogenized multivariate Tutte polynomial of a matroid is Lorentzian whenever the parameter $q$ satisfies $0\lt q \leq 1$. 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 $\mathrm {M}$-matrix is Lorentzian. This refines a result of Holtz who proved that the coefficients of the characteristic polynomial of an $\mathrm {M}$-matrix form an ultra log-concave sequence.

## Authors

Petter Brändén

KTH Royal Institute of Technology, Stockholm, Sweden

June Huh

Institute for Advanced Study and Princeton University, Princeton, NJ, USA, Korea Institute for Advanced Study, Seoul, Korea