Tropical types and associated cellular resolutions

TL;DR

This paper explores how arrangements of tropical hyperplanes induce type data that lead to cellular resolutions of monomial ideals, connecting tropical geometry with combinatorial and algebraic structures, and providing an algorithm for analyzing tropical complexes.

Contribution

It introduces a novel connection between tropical hyperplane arrangements, cellular resolutions, and mixed subdivisions, extending existing constructions and offering an algebraic algorithm for tropical complex analysis.

Findings

Decomposition of tropical torus yields minimal cocellular resolutions.

Supports cellular resolutions on mixed subdivisions of dilated simplices.

Provides an algorithm to compute the facial structure of tropical complexes.

Abstract

An arrangement of finitely many tropical hyperplanes in the tropical torus leads to a notion of `type' data for points, with the underlying unlabeled arrangement giving rise to `coarse type'. It is shown that the decomposition of the tropical torus induced by types gives rise to minimal cocellular resolutions of certain associated monomial ideals. Via the Cayley trick from geometric combinatorics this also yields cellular resolutions supported on mixed subdivisions of dilated simplices, extending previously known constructions. Moreover, the methods developed lead to an algebraic algorithm for computing the facial structure of arbitrary tropical complexes from point data.