{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1996:MN7LUBLD4J46SDIMPU4RSC6JHH","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"5d9ca8e982b89dd74f118df56e9225e91d3fb99016314dca244865b4e7be5f95","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"1996-05-02T00:00:00Z","title_canon_sha256":"a10fad2af314193281d1ad7c2873505ea7b317979e798d5fa55b74795b3bc770"},"schema_version":"1.0","source":{"id":"math/9605216","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9605216","created_at":"2026-05-18T01:05:47Z"},{"alias_kind":"arxiv_version","alias_value":"math/9605216v1","created_at":"2026-05-18T01:05:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9605216","created_at":"2026-05-18T01:05:47Z"},{"alias_kind":"pith_short_12","alias_value":"MN7LUBLD4J46","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_16","alias_value":"MN7LUBLD4J46SDIM","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_8","alias_value":"MN7LUBLD","created_at":"2026-05-18T12:25:48Z"}],"graph_snapshots":[{"event_id":"sha256:b4043f8989211dd3c2088ad6ae1869641500149485b1806c68f5dfc97a4740dc","target":"graph","created_at":"2026-05-18T01:05:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In an earlier work, the authors have determined all possible weights $n$ for which there exists a vanishing sum $\\zeta_1+\\cdots +\\zeta_n=0$ of $m$th roots of unity $\\zeta_i$ in characteristic 0. In this paper, the same problem is studied in finite fields of characteristic $p$. For given $m$ and $p$, results are obtained on integers $n_0$ such that all integers $n\\geq n_0$ are in the ``weight set'' $W_p(m)$. The main result $(1.3)$ in this paper guarantees, under suitable conditions, the existence of solutions of $x_1^d+\\cdots+x_n^d=0$ with all coordinates not equal to zero over a finite field.","authors_text":"K. H. Leung, T. Y. Lam","cross_cats":[],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"1996-05-02T00:00:00Z","title":"On vanishing sums of $\\,m\\,$th roots of unity in finite fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9605216","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8e09f12d43982aeca7b482e1ad8aed22bbc78931f463b410c46570d063bff77d","target":"record","created_at":"2026-05-18T01:05:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"5d9ca8e982b89dd74f118df56e9225e91d3fb99016314dca244865b4e7be5f95","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"1996-05-02T00:00:00Z","title_canon_sha256":"a10fad2af314193281d1ad7c2873505ea7b317979e798d5fa55b74795b3bc770"},"schema_version":"1.0","source":{"id":"math/9605216","kind":"arxiv","version":1}},"canonical_sha256":"637eba0563e279e90d0c7d39190bc939f6058a7444359ba832370f0b5eb36c71","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"637eba0563e279e90d0c7d39190bc939f6058a7444359ba832370f0b5eb36c71","first_computed_at":"2026-05-18T01:05:47.468674Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:47.468674Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u50LggZagGiMpkVt/gveHBErw0q9AHSY8yj4Do2ucVzh0RI7BAYILTUcEcysF7T4f95wtyfJx0YRCpK/Rj9SAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:47.469404Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9605216","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8e09f12d43982aeca7b482e1ad8aed22bbc78931f463b410c46570d063bff77d","sha256:b4043f8989211dd3c2088ad6ae1869641500149485b1806c68f5dfc97a4740dc"],"state_sha256":"695be4e44361440ad68b88fe75170f9a4f431f60424db3d880b8375890fe63f8"}