$\mathcal{K}$-Lorentzian Polynomials

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.