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We give a recursive formulation for the lengths of the k[X]-module k[X]/(I_2(X) + (x_{1,1}^q,..., x_{m,n}^q)) as q varies over all positive integers using Grobner basis. This is a generalized Hilbert-Kunz function, and our formulation proves that it is a polynomial function in q. We give closed forms for the cases when m is at most 2, %as well as the closed forms "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1206.1015","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-06-05T18:25:53Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"113461acc2a54a2c80d91a7b900bb79be02479b98008c16f33056d33ffa783d5","abstract_canon_sha256":"833a89f4e8b7adef5a62d43737de77d6753f70d22009f76af2440164de2e6ba4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:43:27.066037Z","signature_b64":"HRMCyVolZnQVQEXrzy65JsNdpbwDy5J06Rkl0/ws9CmeciPtrXFqeC6enNdDuddKKHwX2rtXsrdFj9h2jSFXAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c3ec5986f9e3be1fe630cefb11309093252634be43ac31f948f0b5ea962585c8","last_reissued_at":"2026-05-18T03:43:27.065530Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:43:27.065530Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hilbert-Kunz functions of 2 x 2 determinantal rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Irena Swanson, Lance Edward Miller","submitted_at":"2012-06-05T18:25:53Z","abstract_excerpt":"Let k be an arbitrary field (of arbitrary characteristic) and let X = [x_{i,j}] be a generic m x n matrix of variables. 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