{"paper":{"title":"The Leray Dimension of a Convex Code","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Carina Curto, Ram\\'on Vera","submitted_at":"2016-12-22T20:58:12Z","abstract_excerpt":"Convex codes were recently introduced as models for neural codes in the brain. Any convex code $\\C$ has an associated minimal embedding dimension $d(\\C)$, which is the minimal Euclidean space dimension such that the code can be realized by a collection of convex open sets. In this work we import tools from combinatorial commutative algebra in order to obtain better bounds on $d(\\C)$ from an associated simplicial complex $\\Delta(\\C)$. In particular, we make a connection to minimal free resolutions of Stanley-Reisner ideals, and observe that they contain topological information that provides str"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.07797","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":""},"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"}