{"paper":{"title":"On zero-sum polytopes: reciprocity, rigidity, and cyclic sieving","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dongchun Han, Hanbin Zhang, Shiwen Zhang, Xuan Wang","submitted_at":"2026-06-09T15:39:54Z","abstract_excerpt":"Let $G$ be a finite abelian group of order $n$, and let $\\mathsf M(G,m)$ denote the set of zero-sum sequences over $G$ of length $m$. We introduce the zero-sum polytope $\\mathcal P_G$, a rational polytope of dimension $n-1$, whose lattice points encode zero-sum sequences: \\[\n  |\\mathsf M(G,m)|=|m\\mathcal P_G\\cap \\mathbb Z^n|. \\] This naturally realizes the enumeration of zero-sum sequences as a problem in rational Ehrhart theory, which leads to a combinatorial reciprocity theorem identifying the negative evaluations of the corresponding counting quasipolynomial with zero-sum sequences of full "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11005","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/2606.11005/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"}