{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:I2FFVJV4MSDHL5TQA4MY24YVUJ","short_pith_number":"pith:I2FFVJV4","schema_version":"1.0","canonical_sha256":"468a5aa6bc648675f67007198d7315a24999a118e0fc816a4e9345bf3b514cb7","source":{"kind":"arxiv","id":"2212.12396","version":3},"attestation_state":"computed","paper":{"title":"Waring numbers over finite commutative local rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AC","authors_text":"Denis E. Videla, Ricardo A. Podest\\'a","submitted_at":"2022-12-23T15:27:22Z","abstract_excerpt":"In this paper we study Waring numbers $g_R(k)$ for $(R,\\frak m)$ a finite commutative local ring with identity and $k \\in \\mathbb{N}$ with $(k,|R|)=1$. We first relate the Waring number $g_R(k)$ with the diameter of the Cayley graphs $G_R(k)=Cay(R,U_R(k))$ and $W_R(k)=Cay(R,S_R(k))$ with $U_R(k) = \\{ x^k : x\\in R^*\\}$ and $S_R(k)=\\{x^k : x\\in R^\\times\\}$, distinguishing the cases where the graphs are directed or undirected. We show that in both cases (directed or undirected), the graph $G_R(k)$ can be obtained by blowing-up the vertices of $G_{\\mathbb{F}_q}(k)$ a number $|\\frak{m}|$ of times, "},"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":"2212.12396","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2022-12-23T15:27:22Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"0e8af3f25123af65b4a840df18426f6296d928173d82c5405b57520bd14a6fb8","abstract_canon_sha256":"4f8c8dab50d1229e5253d1e7a5df4afcc02c81802a90f058ec5c0aca71ac4e8a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:20:39.671654Z","signature_b64":"n79XcGDOBuJtF/sKItLNAqck/iVhPrVk/lZuD8chyWbnXNRobs2SR0SeH1NSiPY8q3r+CTEmgZzneM7D1tUPCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"468a5aa6bc648675f67007198d7315a24999a118e0fc816a4e9345bf3b514cb7","last_reissued_at":"2026-07-05T06:20:39.671169Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:20:39.671169Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Waring numbers over finite commutative local rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AC","authors_text":"Denis E. Videla, Ricardo A. Podest\\'a","submitted_at":"2022-12-23T15:27:22Z","abstract_excerpt":"In this paper we study Waring numbers $g_R(k)$ for $(R,\\frak m)$ a finite commutative local ring with identity and $k \\in \\mathbb{N}$ with $(k,|R|)=1$. We first relate the Waring number $g_R(k)$ with the diameter of the Cayley graphs $G_R(k)=Cay(R,U_R(k))$ and $W_R(k)=Cay(R,S_R(k))$ with $U_R(k) = \\{ x^k : x\\in R^*\\}$ and $S_R(k)=\\{x^k : x\\in R^\\times\\}$, distinguishing the cases where the graphs are directed or undirected. We show that in both cases (directed or undirected), the graph $G_R(k)$ can be obtained by blowing-up the vertices of $G_{\\mathbb{F}_q}(k)$ a number $|\\frak{m}|$ of times, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2212.12396","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2212.12396/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2212.12396","created_at":"2026-07-05T06:20:39.671225+00:00"},{"alias_kind":"arxiv_version","alias_value":"2212.12396v3","created_at":"2026-07-05T06:20:39.671225+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2212.12396","created_at":"2026-07-05T06:20:39.671225+00:00"},{"alias_kind":"pith_short_12","alias_value":"I2FFVJV4MSDH","created_at":"2026-07-05T06:20:39.671225+00:00"},{"alias_kind":"pith_short_16","alias_value":"I2FFVJV4MSDHL5TQ","created_at":"2026-07-05T06:20:39.671225+00:00"},{"alias_kind":"pith_short_8","alias_value":"I2FFVJV4","created_at":"2026-07-05T06:20:39.671225+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/I2FFVJV4MSDHL5TQA4MY24YVUJ","json":"https://pith.science/pith/I2FFVJV4MSDHL5TQA4MY24YVUJ.json","graph_json":"https://pith.science/api/pith-number/I2FFVJV4MSDHL5TQA4MY24YVUJ/graph.json","events_json":"https://pith.science/api/pith-number/I2FFVJV4MSDHL5TQA4MY24YVUJ/events.json","paper":"https://pith.science/paper/I2FFVJV4"},"agent_actions":{"view_html":"https://pith.science/pith/I2FFVJV4MSDHL5TQA4MY24YVUJ","download_json":"https://pith.science/pith/I2FFVJV4MSDHL5TQA4MY24YVUJ.json","view_paper":"https://pith.science/paper/I2FFVJV4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2212.12396&json=true","fetch_graph":"https://pith.science/api/pith-number/I2FFVJV4MSDHL5TQA4MY24YVUJ/graph.json","fetch_events":"https://pith.science/api/pith-number/I2FFVJV4MSDHL5TQA4MY24YVUJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I2FFVJV4MSDHL5TQA4MY24YVUJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I2FFVJV4MSDHL5TQA4MY24YVUJ/action/storage_attestation","attest_author":"https://pith.science/pith/I2FFVJV4MSDHL5TQA4MY24YVUJ/action/author_attestation","sign_citation":"https://pith.science/pith/I2FFVJV4MSDHL5TQA4MY24YVUJ/action/citation_signature","submit_replication":"https://pith.science/pith/I2FFVJV4MSDHL5TQA4MY24YVUJ/action/replication_record"}},"created_at":"2026-07-05T06:20:39.671225+00:00","updated_at":"2026-07-05T06:20:39.671225+00:00"}