Tropical Types and Associated Cellular Resolutions

An arrangement of finitely many tropical hyperplanes in the tropical torus T leads to a notion of ‘type’ data for points in T, with the underlying unlabeled arrangement giving rise to ‘coarse type’. It is shown that the decomposition of T 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.