A presentation of general multipersistence modules computable in polynomial time?

TL;DR

This paper introduces a new presentation of multipersistence homology modules enabling polynomial-time computation of their Groebner bases, improving efficiency over previous methods that could be exponential.

Contribution

The authors present a novel presentation of multipersistence modules that allows for polynomial-time algorithms, avoiding the exponential complexity of prior approaches.

Findings

Polynomial-time algorithm for Groebner bases of multipersistence modules

Avoidance of the mapping telescope in computations

Enhanced efficiency in multipersistence homology calculations

Abstract

Multipersistence homology modules were introduced by G.Carlsson and A.Zomorodian which gave, together with G.Singh, an algorithm to compute their Groebner bases. Although their algorithm has polynomial complexity when the chain modules are free, i.e. in the one-critical case, it might be exponential in general. We give a new presentation of multipersistence homology modules, which allows us to design an algorithm to compute their Groebner bases always in polynomial time by avoiding the mapping telescope.