The number of extreme points of tropical polyhedra

TL;DR

This paper extends McMullen's upper bound theorem to tropical polyhedra, establishing that the maximum number of extreme points matches the classical bound and providing explicit bounds through combinatorial constructions.

Contribution

It demonstrates the validity of McMullen's bound in the tropical setting and introduces lattice path methods to analyze extreme points of tropical polars.

Findings

The upper bound on extreme points holds in tropical polyhedra.

Explicit bounds are derived using lattice path bijections.

Asymptotic tightness of bounds as dimension grows.

Abstract

The celebrated upper bound theorem of McMullen determines the maximal number of extreme points of a polyhedron in terms of its dimension and the number of constraints which define it, showing that the maximum is attained by the polar of the cyclic polytope. We show that the same bound is valid in the tropical setting, up to a trivial modification. Then, we study the natural candidates to be the maximizing polyhedra, which are the polars of a family of cyclic polytopes equipped with a sign pattern. We construct bijections between the extreme points of these polars and lattice paths depending on the sign pattern, from which we deduce explicit bounds for the number of extreme points, showing in particular that the upper bound is asymptotically tight as the dimension tends to infinity, keeping the number of constraints fixed. When transposed to the classical case, the previous constructions yield some lattice path generalizations of Gale's evenness criterion.