$\mathcal{K}$-Lorentzian Polynomials

TL;DR

This paper extends the concept of Lorentzian polynomials to proper convex cones, exploring their properties, connections to positive linear maps, and the complexity of understanding quadratic cases, revealing deep structural insights.

Contribution

It introduces $ ext{K}$-Lorentzian polynomials on convex cones, linking them to $ ext{K}$-positive linear maps and analyzing their properties and computational complexity.

Findings

$ ext{K}$-Lorentzian polynomials relate to $ ext{K}$-positive linear maps.

Quadratic $ ext{K}$-Lorentzian polynomials are computationally complex.

$ ext{K}$-Lorentzian and $ ext{K}$-completely log-concave polynomials coincide.

Abstract

Lorentzian polynomials are a fascinating class of real polynomials with many applications. Their definition is specific to the nonnegative orthant. Following recent work, we examine Lorentzian polynomials on proper convex cones. For a self-dual cone $\mathcal{K}$ we find a connection between $\mathcal{K}$-Lorentzian polynomials and $\mathcal{K}$-positive linear maps, which were studied in the context of the generalized Perron-Frobenius theorem. We find that as the cone $\mathcal{K}$ varies, even the set of quadratic $\mathcal{K}$-Lorentzian polynomials can be difficult to understand algorithmically. We also show that, just as in the case of the nonnegative orthant, $\mathcal{K}$-Lorentzian and $\mathcal{K}$-completely log-concave polynomials coincide.