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Here we prove (N_r) provided $r=k^2+\\alpha,1\\leq\\alpha\\leq2k,k\\geq 3$ and either (i) $\\alpha$ is odd and $\\alpha\\geq \\sqrt{2k}$ or (ii) $\\alpha$ is even and at lest 6, and the fractional part of $\\sqrt{r}$ is at most $2(\\sqrt{2}-1)$."},"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":"math/9809101","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.AG","submitted_at":"1998-09-18T04:12:20Z","cross_cats_sorted":[],"title_canon_sha256":"09d63ba219815f61b10e7aefd0fb282ad9864161f16419c105c71fa6d06cffde","abstract_canon_sha256":"44c8570ab81e56683e7cce57b7e039cac9c90ffafa371ba7cc3879fa85231069"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:10.795780Z","signature_b64":"hB0s/OAAsZ1K8J7rwIiToF2Gvm74H2obR7Ff85u79W6SGQsSjIY0wSTkeiW81PhM5L9KSq8Uu9r9ChYpPqiZCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0836b460ee75109e216dc776ca79b261083921e76e31140c5732d24733a17025","last_reissued_at":"2026-05-18T03:26:10.795174Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:10.795174Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Nagata Problem","license":"","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Ziv Ran","submitted_at":"1998-09-18T04:12:20Z","abstract_excerpt":"Nagata has conjectured that the following statement (N_r) holds for all $r\\geq 10$: (N_r) if $P_1,...P_r \\in {\\mathbb P}^2$ are generic points then any plane curve $C$ satisfies $\\sum_1^r mult_{P_i}(C)\\leq \\sqrt{r} deg(C)$. Nagata proved (N_r) whenever $r$ is a perfect square. Here we prove (N_r) provided $r=k^2+\\alpha,1\\leq\\alpha\\leq2k,k\\geq 3$ and either (i) $\\alpha$ is odd and $\\alpha\\geq \\sqrt{2k}$ or (ii) $\\alpha$ is even and at lest 6, and the fractional part of $\\sqrt{r}$ is at most $2(\\sqrt{2}-1)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9809101","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/9809101","created_at":"2026-05-18T03:26:10.795256+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/9809101v3","created_at":"2026-05-18T03:26:10.795256+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9809101","created_at":"2026-05-18T03:26:10.795256+00:00"},{"alias_kind":"pith_short_12","alias_value":"BA3LIYHOOUIJ","created_at":"2026-05-18T12:25:49.038998+00:00"},{"alias_kind":"pith_short_16","alias_value":"BA3LIYHOOUIJ4ILN","created_at":"2026-05-18T12:25:49.038998+00:00"},{"alias_kind":"pith_short_8","alias_value":"BA3LIYHO","created_at":"2026-05-18T12:25:49.038998+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BA3LIYHOOUIJ4ILNY53MU6NSME","json":"https://pith.science/pith/BA3LIYHOOUIJ4ILNY53MU6NSME.json","graph_json":"https://pith.science/api/pith-number/BA3LIYHOOUIJ4ILNY53MU6NSME/graph.json","events_json":"https://pith.science/api/pith-number/BA3LIYHOOUIJ4ILNY53MU6NSME/events.json","paper":"https://pith.science/paper/BA3LIYHO"},"agent_actions":{"view_html":"https://pith.science/pith/BA3LIYHOOUIJ4ILNY53MU6NSME","download_json":"https://pith.science/pith/BA3LIYHOOUIJ4ILNY53MU6NSME.json","view_paper":"https://pith.science/paper/BA3LIYHO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/9809101&json=true","fetch_graph":"https://pith.science/api/pith-number/BA3LIYHOOUIJ4ILNY53MU6NSME/graph.json","fetch_events":"https://pith.science/api/pith-number/BA3LIYHOOUIJ4ILNY53MU6NSME/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BA3LIYHOOUIJ4ILNY53MU6NSME/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BA3LIYHOOUIJ4ILNY53MU6NSME/action/storage_attestation","attest_author":"https://pith.science/pith/BA3LIYHOOUIJ4ILNY53MU6NSME/action/author_attestation","sign_citation":"https://pith.science/pith/BA3LIYHOOUIJ4ILNY53MU6NSME/action/citation_signature","submit_replication":"https://pith.science/pith/BA3LIYHOOUIJ4ILNY53MU6NSME/action/replication_record"}},"created_at":"2026-05-18T03:26:10.795256+00:00","updated_at":"2026-05-18T03:26:10.795256+00:00"}