Tropicalization of facets of polytopes

TL;DR

This paper establishes a method to represent pure tropical polytopes as intersections of tropical half-spaces derived from lifts, providing explicit constructions and confirming a conjecture for general position cases.

Contribution

It proves that pure tropical polytopes can be described via lifts and tropical half-spaces, explicitly constructing such lifts and confirming a conjecture in general position.

Findings

Pure tropical polytopes are intersections of tropical half-spaces from lifts.

Explicit construction of lifts considering geometric properties.

Confirmation of Develin and Yu's conjecture for polytopes in general position.

Abstract

It is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the Puiseux series field. The latter polytopes are called lifts of the tropical polytope. We prove that any pure tropical polytope is the intersection of the tropical half-spaces given by the images under the valuation map of the facet-defining half-spaces of a certain lift. We construct this lift explicitly, taking into account geometric properties of the given polytope. Moreover, when the generators of the tropical polytope are in general position, we prove that the above property is satisfied for any lift. This solves a conjecture of Develin and Yu.