nLab Lorentzian polynomial

Idea

Lorentzian polynomials are a class of homogeneous polynomials with real coefficients that are used to relate continuous convex analysis and discrete convex analysis via tropical geometry. The theory of Lorentzian polynomials provides techniques to prove that various naturally-occurring sequences of non-negative real numbers, $(a_i)$, are log-concave, i.e., $a_k^2 \geq a_{k-1}a_{k+1}$ for all $k$.

Definition

Definition

Let $f \in \mathbb{R}[x_1, \ldots, x_n]$ be a homogeneous polynomial of degree $d$ such that all coefficients of $f$ are strictly positive. If either $d = 0, 1$, or $d\geq 2$ and the Hessian of any $(d-2)$th partial derivative $\partial_{i_1} \cdots \partial_{i_{d-2}}f$ has Hessian matrix with signature $(+,-,-,\ldots,-)$, then $f$ is said to be strictly Lorentzian. If $f$ is in the closure of the set of strictly Lorentzian polynomials, then it is said to be Lorentzian.

Related concepts

• polynomial
• matroid

References

• Petter Brändén, June Huh, Lorentzian polynomials, Annals of Mathematics, 192 3 (2020) [arXiv:1902.03719, doi:10.4007/annals.2020.192.3.4]

• June Huh, Lorentzian polynomials, Simons Lectures 2019 (videos)

• June Huh, Combinatorics and Hodge theory, Proc. Int. Cong. Math. 1 (2022)

  [doi:10.4171/ICM2022/205, pdf, pdf]