  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.
