Slicing and dicing polytopes

TL;DR

This paper proves that regularity is not necessary for polytopes to support cellular resolutions, introducing a new sufficient condition and employing discrete Morse theory for the proof.

Contribution

It provides a new, easily checked condition for polytopes to support cellular resolutions, removing the need for regularity.

Findings

Regularity condition can be omitted for cellular resolutions

Introduces a new sufficient condition for subdivided polytopes

Uses discrete Morse theory as a key tool

Abstract

Using tropical convexity Dochtermann, Fink, and Sanyal proved that regular fine mixed subdivisions of Minkowski sums of simplices support minimal cellular resolutions. They asked if the regularity condition can be removed. We give an affirmative answer by a different method. A new easily checked sufficient condition for a subdivided polytope to support a cellular resolution is proved. The main tool used is discrete Morse theory.