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

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 Gröbner basis calculations.