Monomial polarization and depolarization of abstract simplicial complexes

TL;DR

This paper introduces a translation of polarization and depolarization operations from monomial ideals to abstract simplicial complexes, enabling new methods for complex reduction and efficient Alexander dual computation.

Contribution

It provides a novel translation of algebraic operations to simplicial complexes and proposes a reduction technique that preserves homology for computational efficiency.

Findings

Explicit relation between Koszul complexes of monomial ideals and their polarizations

Reduction method preserves homology and simplifies complexes

Improved efficiency in computing Alexander duals

Abstract

We translate the operations of polarization and depolarization from monomial ideals in a polynomial ring to abstract simplicial complexes. As a result, we explicitly describe the relation between the Koszul simplicial complex of a monomial ideal and that of its polarization. Using the simplicial translation of depolarization we propose a way to reduce a simplicial complex to a smaller one with the same homology. This type of reduction, that can be interpreted as non-elementary collapse, can be used as a pre-process step for algorithms on simplicial complexes. We apply this methodology to the efficient computation of the Alexander dual of abstract simplicial complexes.