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

Mara Pompili, Daniel Smertnig

TL;DR

The paper develops a unifying computational framework for factorization theory in upper cluster algebras by introducing finite Laurent intersection rings (FLIRs). It proves that locally acyclic cluster algebras are FLIRs (and hence Krull domains), enabling computable class groups, factoriality tests, and explicit factorizations without Gröbner basis computations. Banff-cluster algebras are shown to be explicit FLIRs, and the authors provide algorithms to compute the class group, distributions of height-one primes, and factorization data, with practical performance demonstrated through computational experiments. The framework generalizes to generalized cluster algebras and Laurent phenomenon algebras, and the results yield new structural insights (e.g., Nagata-type decompositions and PDE conditions) while pointing to open questions about the limits of the FLIR approach and the complexity of the Banff algorithm.

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 Gröbner basis calculations.

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

TL;DR

The paper develops a unifying computational framework for factorization theory in upper cluster algebras by introducing finite Laurent intersection rings (FLIRs). It proves that locally acyclic cluster algebras are FLIRs (and hence Krull domains), enabling computable class groups, factoriality tests, and explicit factorizations without Gröbner basis computations. Banff-cluster algebras are shown to be explicit FLIRs, and the authors provide algorithms to compute the class group, distributions of height-one primes, and factorization data, with practical performance demonstrated through computational experiments. The framework generalizes to generalized cluster algebras and Laurent phenomenon algebras, and the results yield new structural insights (e.g., Nagata-type decompositions and PDE conditions) while pointing to open questions about the limits of the FLIR approach and the complexity of the Banff algorithm.

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 Gröbner basis calculations.
Paper Structure (32 sections, 44 theorems, 85 equations, 2 figures, 1 table, 8 algorithms)

This paper contains 32 sections, 44 theorems, 85 equations, 2 figures, 1 table, 8 algorithms.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Factorization
  5. Krull Domains and Class Groups
  6. (Laurent) Polynomial Rings
  7. Divisorial Ideals and Divisors
  8. Class Groups
  9. Cluster Algebras and Seeds
  10. Seeds and Mutations
  11. Cluster Algebras
  12. Locally Acyclic Cluster Algebras
  13. Laurent Intersection Rings
  14. Iterated FLIRs
  15. One-variable LIRs
  16. ...and 17 more sections

Key Result

theorem 1

Let $D$ be a computable Krull domain (such as $\mathbb Q$, $\overline{\mathbb Q}$, a finite field, or $\mathbb Z$), and $\mathbb P$ a computable semifield (such as a tropical semifield). Let $(\bm{x}, \bm{y}, B)$ be a seed with the exchange matrix $B$ satisfying the Banff property, and let $A=A(\bm{

Figures (2)

  • Figure 1: A triangulation of $D_{2,2}$.
  • Figure 2: Recursive construction of triangulations of $D_{m+1,p}$ and $D_{m,p+1}$ from $D_{m,p}$.

Theorems & Definitions (107)

  • theorem 1
  • theorem 2
  • definition 1
  • definition 2
  • remark 1
  • theorem 3: FOSSUM
  • theorem 4: Nagata's Theorem FOSSUM
  • corollary 1: FOSSUM or gilmer84
  • remark 2
  • definition 3
  • ...and 97 more