{"paper":{"title":"Gr\\\"obner Cones for Finite Type Cluster Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Karolyn So, Nathan Ilten","submitted_at":"2025-01-13T05:22:45Z","abstract_excerpt":"Let $\\mathcal{A}$ be a cluster algebra of finite cluster type. We study the Gr\\\"obner cone $\\mathcal{C}_{\\mathcal{A}}$ parametrizing term orders inducing an initial degeneration of the ideal $I_{\\mathcal{A}}$ of relations among the cluster variables of $\\mathcal{A}$ to the ideal generated by products of incompatible cluster variables. We show that for any cluster variable $v$, the weight induced by taking compatibility degrees with $v$ belongs to $\\mathcal{C}_{\\mathcal{A}}$. This allows us to construct an explicit circular term order and prove a conjecture of Ilten, N\\'ajera Ch\\'avez, and Tref"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.07065","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.07065/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}