Factoriality and Class Groups of Upper Cluster Algebras and Finite Laurent Intersection Rings: A Computational Approach

TL;DR

This paper introduces algorithms for computing class groups and factoriality of a new class of rings called finite Laurent intersection rings, which include various cluster algebras, using polynomial factorizations instead of Gr"obner bases.

Contribution

It presents a computational framework for analyzing factoriality and class groups of FLIRs, expanding tools for cluster algebra research.

Findings

Algorithms for class group computation of FLIRs

Methods to determine factoriality of these rings

Procedures to find all factorizations of elements

Abstract

We study factoriality and the class groups of locally acyclic cluster algebras. To do so, we introduce a new class of rings called finite Laurent intersection rings (FLIRs), which includes locally acyclic cluster algebras, full-rank upper cluster algebras, and certain generalized upper cluster algebras and Laurent phenomenon algebras. Our main results are algorithms to compute the class group of an explicit FLIR, to determine factoriality, and to compute all factorizations of a given element. The algorithms are based on multivariate polynomial factorizations, avoiding computationally expensive Gr\"obner basis calculations.