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Necessary and sufficient conditions are given for $\\ov{I^{r+1}J^{s+1}}=a\\ov{I^rJ^{s+1}}+b\\ov{I^{r+1}J^s}$ to hold ${for all}r \\geq r_0{and}s \\geq s_0$ in terms of vanishing of $[H^2_{(at_1,bt_2)}(\\ov{\\mathcal{R}^\\prime}(I,J))]_{(r_0,s_0)}$, where $a \\in I,b \\in J$ is a good joint reduction of the filtration $\\{\\ov{I^rJ^s}\\}.$ This is used to derive a theorem due to Rees on normal joint reduction number zero. The vanishing o"},"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":"1307.3431","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-07-12T12:20:36Z","cross_cats_sorted":[],"title_canon_sha256":"dfaad6249f000681f18e5d3227ddf7329b24f22d372fa458a6ac5d6c2b549a85","abstract_canon_sha256":"aa1d5e46a56eb0d98554dcb4d80821f307d14311073c8c3aa168d6e9f8c76fb3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:39.176179Z","signature_b64":"qAbiI4zmtiMKU6iYDi19uRCR0ITIZzSm2RQtUiTUy0q5Dyx7/3PmTPslJiQzrEpbRDci4588Yb3NK8CMo4CVBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"079849127aa35c517167d29ad36f746177c04359c3415ec0784cea07b8587d4f","last_reissued_at":"2026-05-18T03:18:39.175616Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:39.175616Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Local Cohomology of Bigraded Rees Algebras and Normal Hilbert Coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"J. K. Verma, Shreedevi K. Masuti","submitted_at":"2013-07-12T12:20:36Z","abstract_excerpt":"Let $(R,\\m)$ be an analytically unramified Cohen-Macaulay local ring of dimension 2 with infinite residue field and $\\ov{I}$ be the integral closure of an ideal $I$ in $R$. Necessary and sufficient conditions are given for $\\ov{I^{r+1}J^{s+1}}=a\\ov{I^rJ^{s+1}}+b\\ov{I^{r+1}J^s}$ to hold ${for all}r \\geq r_0{and}s \\geq s_0$ in terms of vanishing of $[H^2_{(at_1,bt_2)}(\\ov{\\mathcal{R}^\\prime}(I,J))]_{(r_0,s_0)}$, where $a \\in I,b \\in J$ is a good joint reduction of the filtration $\\{\\ov{I^rJ^s}\\}.$ This is used to derive a theorem due to Rees on normal joint reduction number zero. 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