Lorentzian polynomials from polytope projections

TL;DR

This paper explores Lorentzian polynomials derived from projections of flow polytopes, extending previous work on generalized permutahedra and revealing new Lorentzian properties through polytopal transformations.

Contribution

It demonstrates that certain projections of integer point transforms of flow polytopes are Lorentzian, broadening the class of polynomials with this property.

Findings

Projections of flow polytope transforms are Lorentzian.

Extends Lorentzian polynomial theory to flow polytopes.

Connects polytopal projections with log-concavity properties.

Abstract

Lorentzian polynomials, recently introduced by Br\"and\'en and Huh, generalize the notion of log-concavity of sequences to homogeneous polynomials whose supports are integer points of generalized permutahedra. Br\"and\'en and Huh show that normalizations of polynomials equaling 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 (which, before projection, are not Lorentzian), are also Lorentzian.