Polynomials with Lorentzian Signature, and Computing Permanents via Hyperbolic Programming

TL;DR

This paper introduces a new class of polynomials with Lorentzian signature, generalizes Lorentzian polynomials, and develops a hyperbolic programming method for computing permanents of certain matrices.

Contribution

It extends the class of Lorentzian polynomials to include hyperbolic and conic stable polynomials, and proposes a novel approach for permanent computation using hyperbolic programming.

Findings

Hyperbolic and conic stable polynomials are included in the Lorentzian signature class.

The set of polynomials with Lorentzian signature is shown to be closed.

A new method for computing permanents of specific matrices via hyperbolic programming is developed.

Abstract

We study the class of polynomials whose Hessians evaluated at any point of a closed convex cone have Lorentzian signature. This class is a generalization to the remarkable class of Lorentzian polynomials. We prove that hyperbolic polynomials and conic stable polynomials belong to this class, and the set of polynomials with Lorentzian signature is closed. Finally, we develop a method for computing permanents of nonsingular matrices which belong to a class that includes nonsingular $k$-locally singular matrices via hyperbolic programming.