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. 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 $0 < q \le 1$. Consequences are proofs of the strongest Mason's conjecture from 1972 and negative dependence properties of the random cluster model 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.

中文翻译：

---

洛伦兹多项式

我们研究洛伦兹多项式的类。该类包含齐次稳定多项式以及凸体和射影簇的体积多项式。我们证明非零洛伦兹多项式的 Hessian 在正数上的任何一点都恰好有一个正特征值。这个性质可以看作是洛伦兹多项式的霍奇-黎曼关系的类比。洛伦兹多项式与拟阵理论和负相关特性密切相关。我们证明拟阵，更普遍的是 M 凸集，具有洛伦兹性质的特征，并围绕洛伦兹多项式发展了一个理论。特别是，我们提供了一大类线性算子，它们保留了洛伦兹性质，并证明了洛伦兹测度具有几个负相关性。我们还证明了热带化洛伦兹多项式的类与离散凸分析意义上的 M 凸函数类是一致的。热带连接用于从 M 凸函数生成洛伦兹多项式。我们给出一般理论的两个应用。首先，我们证明当参数 q 满足 $0 < q \le 1$ 时，拟阵的均质化多元 Tutte 多项式是洛伦兹多项式。结果证明了 1972 年以来最强的梅森猜想和统计物理学中随机聚类模型模型的负相关特性。其次，我们证明了 M 矩阵的多元特征多项式是洛伦兹的。这改进了 Holtz 的结果，他证明了 M 矩阵的特征多项式的系数形成了超对数凹序列。