Lorentzian polynomials from polytope projections

Abstract

Lorentzian polynomials, recently introduced by Brändén and Huh, generalize the notion of log-concavity of sequences to homogeneous polynomials whose supports are integer points of generalized permutahedra. Brändén and Huh show that normalizations of integer point transforms of generalized permutahedra are Lorentzian. Moreover, normalizations of certain projections of integer point transforms of generalized permutahedra with zero-one vertices are also Lorentzian. Taking this polytopal perspective further, we show that normalizations of certain projections of integer point transforms of flow polytopes are Lorentzian.

Properties of conically stable polynomials and imaginary projections

“…Recently, due to the rich structure of Lorentzian polynomials, several new connections were established [27,38,81]. Furthermore, Brändén and Leake introduced an even further generalization by studying conically Lorentzian polynomials [28].…”