{"paper":{"title":"Region level via centralization for hyperplane arrangements and beyond","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Finn Southerland, Lani Southern, Su Zhou","submitted_at":"2025-11-12T19:01:36Z","abstract_excerpt":"In \"Faces of a Hyperplane Arrangement Enumerated by Ideal Dimension, with Applications to Plane, Plaids, and Shi,\" Zaslavsky showed how to compute the number $r_\\ell(\\mathcal{A})$ of regions of a real hyperplane arrangement $\\mathcal{A}$ with a given level, refining his well known enumeration of regions and relatively bounded regions. We restate this theorem in terms of a construction called the centralization of $\\mathcal{A}$, give a bijective proof, and then apply it in two ways to answer questions concerning the concept of level. Firstly, a consequence of this enumeration is that $r_\\ell(\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2511.09653","kind":"arxiv","version":2},"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/2511.09653/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"}