Shellings of Tropical Hypersurfaces

TL;DR

This paper proves that tropical hypersurfaces are shellable by examining the shellability of their boundary complexes through compactification and duality, revealing intricate relationships with discrete Morse theory.

Contribution

It establishes the shellability of tropical hypersurfaces and explores the duality and collapsibility properties of associated polyhedral complexes.

Findings

Tropical hypersurfaces are shellable.

The boundary complex of an unbounded polyhedron can be shellable after compactification.

The tight span of a regular subdivision is collapsible but not shellable.

Abstract

The shellability of the boundary complex of an unbounded polyhedron is investigated. To this end, it is necessary to pass to a suitable compactification, e.g., by one point. This observation can be exploited to prove that any tropical hypersurface is shellable. Under the hood there is a subtle interplay between the duality of polyhedral complexes and their shellability. Translated into discrete Morse theory, that interplay entails that the tight span of an arbitrary regular subdivision is collapsible, but not shellable in general.