Asymptotics of the number of lattice points in the transportation polytope via optimization on Lorentzian polynomials.

Abstract

We formally extend the theory of polynomial capacity to power series and totally uni- modular matrices. Using these results, we prove the log-asymptotic correctness of bounds by Brändén, Leake, and Pak developed through the use of Lorentzian polynomials ([BLP23]) under certain conditions, and provide a counterexample where these bounds are not log- asymptotically correct, even when symmetry exists.