{"paper":{"title":"Defect Antichains and Multigraded Symbolic Defect Series of Edge Ideals under Graph Blow-ups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Tabinda Rasheed, Wang Yao","submitted_at":"2026-06-29T05:51:07Z","abstract_excerpt":"In this paper, we study symbolic defect functions of edge ideals through finite antichains of exponent vectors. Let $G$ be a finite simple graph and let $I(G)$ be its edge ideal. For each symbolic degree $s$, we define the symbolic exponent region $\\mathcal{P}_s(G)$, the ordinary exponent region $\\mathcal{O}_s(G)$, and the symbolic defect antichain $\\mathcal{D}_s(G)=\\min\\big(\\mathcal{P}_s(G)\\setminus \\mathcal{O}_s(G)\\big)$, where the minimum is taken with respect to the componentwise partial order. We prove that $\\mathcal{D}_s(G)$ gives a finite obstruction set controlling the minimal monomial"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29817","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/2606.29817/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"}