nLab Lorentzian polynomial

Contents

Contents

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)(a_i), are log-concave, i.e., a k 2a k1a k+1a_k^2 \geq a_{k-1}a_{k+1} for all kk.

Definition

Definition

Let f[x 1,,x n]f \in \mathbb{R}[x_1, \ldots, x_n] be a homogeneous polynomial of degree dd such that all coefficients of ff are strictly positive. If either d=0,1d = 0, 1, or d2d\geq 2 and the Hessian of any (d2)(d-2)th partial derivative i 1 i d2f\partial_{i_1} \cdots \partial_{i_{d-2}}f has Hessian matrix with signature (+,,,,)(+,-,-,\ldots,-), then ff is said to be strictly Lorentzian. If ff is in the closure of the set of strictly Lorentzian polynomials, then it is said to be Lorentzian.

References

Last revised on August 21, 2024 at 01:56:00. See the history of this page for a list of all contributions to it.