{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2007:BREZFBRHXEKJ2F5VNXWZE7PINO","short_pith_number":"pith:BREZFBRH","schema_version":"1.0","canonical_sha256":"0c49928627b9149d17b56ded927de86b85b268d0c0b920a5f1b75c5ffad0b3a2","source":{"kind":"arxiv","id":"math/0703180","version":3},"attestation_state":"computed","paper":{"title":"On value sets of polynomials over a field","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2007-03-07T17:26:21Z","abstract_excerpt":"Let F be any field. Let p(F) be the characteristic of F if F is not of characteristic zero, and let p(F)=+\\infty otherwise. Let A_1,...,A_n be finite nonempty subsets of F, and let\n  $$f(x_1,...,x_n)=a_1x_1^k+...+a_nx_n^k+g(x_1,...,x_n)\\in F[x_1,...,x_n]$$ with k in {1,2,3,...}, a_1,...,a_n in F\\{0} and deg(g)<k. We show that $$|{f(x_1,...,x_n):x_1 in A_1,...,x_n in A_n}| \\geq min{p(F),\\sum_{i=1}^n[(|A_i|-1)/k]+1}.$$ When $k\\geq n$ and $|A_i|\\geq i$ for $i=1,...,n$, we also have $$|{f(x_1,...,x_n):x_1 in A_1,...,x_n in A_n, and x_i not=x_j if i not=j}| \\geq min{p(F),\\sum_{i=1}^n[(|A_i|-i)/k]+1"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0703180","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2007-03-07T17:26:21Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"6dd72628d2059b62edff894961f83f164e58cd72d4830c201ebaf30b6519acb7","abstract_canon_sha256":"bc04f116e623d245f032ea386275292fcede93d6da58b6f820cb22157febeda2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:10:34.046525Z","signature_b64":"kc/RTn9Wmoo0HPnXUeLAoZeWZR+NVWW4pG9cN5PurEfNLhe2IBjVS4FmixcXQN9AYXCjjdKehgzyGpAIqtrnDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c49928627b9149d17b56ded927de86b85b268d0c0b920a5f1b75c5ffad0b3a2","last_reissued_at":"2026-07-04T15:10:34.046118Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:10:34.046118Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On value sets of polynomials over a field","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2007-03-07T17:26:21Z","abstract_excerpt":"Let F be any field. Let p(F) be the characteristic of F if F is not of characteristic zero, and let p(F)=+\\infty otherwise. Let A_1,...,A_n be finite nonempty subsets of F, and let\n  $$f(x_1,...,x_n)=a_1x_1^k+...+a_nx_n^k+g(x_1,...,x_n)\\in F[x_1,...,x_n]$$ with k in {1,2,3,...}, a_1,...,a_n in F\\{0} and deg(g)<k. We show that $$|{f(x_1,...,x_n):x_1 in A_1,...,x_n in A_n}| \\geq min{p(F),\\sum_{i=1}^n[(|A_i|-1)/k]+1}.$$ When $k\\geq n$ and $|A_i|\\geq i$ for $i=1,...,n$, we also have $$|{f(x_1,...,x_n):x_1 in A_1,...,x_n in A_n, and x_i not=x_j if i not=j}| \\geq min{p(F),\\sum_{i=1}^n[(|A_i|-i)/k]+1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0703180","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/math/0703180/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":"math/0703180","created_at":"2026-07-04T15:10:34.046182+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0703180v3","created_at":"2026-07-04T15:10:34.046182+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0703180","created_at":"2026-07-04T15:10:34.046182+00:00"},{"alias_kind":"pith_short_12","alias_value":"BREZFBRHXEKJ","created_at":"2026-07-04T15:10:34.046182+00:00"},{"alias_kind":"pith_short_16","alias_value":"BREZFBRHXEKJ2F5V","created_at":"2026-07-04T15:10:34.046182+00:00"},{"alias_kind":"pith_short_8","alias_value":"BREZFBRH","created_at":"2026-07-04T15:10:34.046182+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/BREZFBRHXEKJ2F5VNXWZE7PINO","json":"https://pith.science/pith/BREZFBRHXEKJ2F5VNXWZE7PINO.json","graph_json":"https://pith.science/api/pith-number/BREZFBRHXEKJ2F5VNXWZE7PINO/graph.json","events_json":"https://pith.science/api/pith-number/BREZFBRHXEKJ2F5VNXWZE7PINO/events.json","paper":"https://pith.science/paper/BREZFBRH"},"agent_actions":{"view_html":"https://pith.science/pith/BREZFBRHXEKJ2F5VNXWZE7PINO","download_json":"https://pith.science/pith/BREZFBRHXEKJ2F5VNXWZE7PINO.json","view_paper":"https://pith.science/paper/BREZFBRH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0703180&json=true","fetch_graph":"https://pith.science/api/pith-number/BREZFBRHXEKJ2F5VNXWZE7PINO/graph.json","fetch_events":"https://pith.science/api/pith-number/BREZFBRHXEKJ2F5VNXWZE7PINO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BREZFBRHXEKJ2F5VNXWZE7PINO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BREZFBRHXEKJ2F5VNXWZE7PINO/action/storage_attestation","attest_author":"https://pith.science/pith/BREZFBRHXEKJ2F5VNXWZE7PINO/action/author_attestation","sign_citation":"https://pith.science/pith/BREZFBRHXEKJ2F5VNXWZE7PINO/action/citation_signature","submit_replication":"https://pith.science/pith/BREZFBRHXEKJ2F5VNXWZE7PINO/action/replication_record"}},"created_at":"2026-07-04T15:10:34.046182+00:00","updated_at":"2026-07-04T15:10:34.046182+00:00"}