symmetric monoidal (∞,1)-category of spectra
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 $k$.
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.
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)
Last revised on August 9, 2022 at 12:18:54. See the history of this page for a list of all contributions to it.