Polynomials with Lorentzian Signature and computing Permanents

Abstract

We study the class of polynomials whose Hessians evaluated at any interior point of a closed convex cone K 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. 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.