Mara Pompili: Computing Factorizations via Laurent Intersection Rings

Computing Factorizations via Laurent Intersection Rings

Mara Pompili, University of Graz

Abstract: Factorization theory in integral domains is influenced by the structure of the divisor class group, yet explicit computations are typically limited to well-understood settings such as number fields or polynomial rings. In this talk, we introduce Laurent intersection rings (LIRs), a class of subrings of rational function fields obtained as intersections of Laurent polynomial rings arising from different coordinate systems. This framework is motivated by the study of cluster algebras, where such intersections naturally occur through the Laurent phenomenon. We focus in particular on finite Laurent intersection rings (FLIRs), which often coincide with Krull domains in this setting. For these rings, we provide an explicit description of the class group in terms of valuations at height-one primes, leading to effective algorithms for computing divisor class groups and factorizations. These methods apply in particular to large classes of cluster algebras. Finally, we discuss the implementation of these algorithms and highlight open questions.

The talk will be live and streamed.

Roman Drnovšek and Daniel Smertnig