K-Lorentzian Polynomials

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 K we find a connection between K-Lorentzian polynomials and K-positive linear maps, which were studied in the context of the generalized Perron-Frobenius theorem. We find that as the cone K varies, even the set of quadratic K-Lorentzian polynomials can be difficult to understand algorithmically. We also show that, just as in the case of the nonnegative orthant, K-Lorentzian and K-completely log-concave polynomials coincide.