{"paper":{"title":"Explicit Hilbert-Kunz functions of 2 x 2 determinantal rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Irena Swanson, Marcus Robinson","submitted_at":"2013-04-26T20:02:48Z","abstract_excerpt":"Let $k[X] = k[x_{i,j}: i = 1,..., m; j = 1,..., n]$ be the polynomial ring in $m n$ variables $x_{i,j}$ over a field $k$ of arbitrary characteristic. Denote by $I_2(X)$ the ideal generated by the $2 \\times 2$ minors of the generic $m \\times n$ matrix $[x_{i,j}]$. We give a closed formulation for the dimensions of the $k$-vector space $k[X]/(I_2(X) + (x_{1,1}^q,..., x_{m,n}^q))$ as $q$ varies over all positive integers, i.e., we give a closed form for the generalized Hilbert-Kunz function of the determinantal ring $k[X]/I_{2}[X]$. We also give a closed formulation of dimensions of related quoti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.7274","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"}