{"paper":{"title":"Lattice-Supported Splines on Polytopal Complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Michael DiPasquale","submitted_at":"2013-12-11T20:00:53Z","abstract_excerpt":"We study the module $C^r(\\mathcal{P})$ of piecewise polynomial functions of smoothness $r$ on a pure $n$-dimensional polytopal complex $\\mathcal{P}\\subset\\mathbb{R}^n$, via an analysis of certain subcomplexes $\\mathcal{P}_W$ obtained from the intersection lattice of the interior codimension one faces of $\\mathcal{P}$. We obtain two main results: first, we show that in sufficiently high degree, the vector space $C^r_k(\\mathcal{P})$ of splines of degree $\\leq k$ has a basis consisting of splines supported on the $\\mathcal{P}_W$ for $k\\gg0$. We call such splines lattice-supported. This shows that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3294","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":""},"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"}