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Fix a monic polynomial $u(x)=x^n +u_{n-1}x^{n-1}+\\cdots +u_{\\ell+1}x^{\\ell+1} \\in \\mathbb{F}_q[x]$ of degree $n$ and consider all degree $n$ monic polynomials of the form $$f(x) = u(x) + v_\\ell(x), \\ v_\\ell(x)=a_\\ell x^\\ell+a_{\\ell-1}x^{\\ell-1}+\\cdots+a_1x+a_0\\in \\mathbb{F}_q[x].$$ For integer $0\\leq k \\leq {\\rm min}\\{n,q\\}$, let $N_k(u(x),\\ell)$ denote the total number of $v_\\ell(x)$ such that $u(x)+v_\\ell(x)$ has exactly $k$ distinct roots in $\\"},"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":"1702.02327","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-02-08T08:50:46Z","cross_cats_sorted":[],"title_canon_sha256":"ae73c57f5af9b5b0db06fb3f960217a1af97891a4e5a0b49ad8bf634fe41090d","abstract_canon_sha256":"75f700201c84fd02c306f329138d3b501c0f36d12aeeb546b2b62a22f559781a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:06.211684Z","signature_b64":"WsKxWaOThTjLSsuA/D+4rfAQJMnv1mNnMkAgUyQofTjIRk6xxuT4Sdj0TVledoRlSZlTohpChiVoLAfudRYsAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"195f604ce2dd932d0f9c331226c282ece59a587955cfd4ebec170933cbf0e4f0","last_reissued_at":"2026-05-18T00:51:06.211311Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:06.211311Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Counting Polynomials with Distinct Zeros in Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Haiyan Zhou, Li-Ping Wang, Weiqiong Wang","submitted_at":"2017-02-08T08:50:46Z","abstract_excerpt":"Let $\\mathbb{F}_q$ be a finite field with $q=p^e$ elements, where $p$ is a prime and $e\\geq 1$ is an integer. 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Fix a monic polynomial $u(x)=x^n +u_{n-1}x^{n-1}+\\cdots +u_{\\ell+1}x^{\\ell+1} \\in \\mathbb{F}_q[x]$ of degree $n$ and consider all degree $n$ monic polynomials of the form $$f(x) = u(x) + v_\\ell(x), \\ v_\\ell(x)=a_\\ell x^\\ell+a_{\\ell-1}x^{\\ell-1}+\\cdots+a_1x+a_0\\in \\mathbb{F}_q[x].$$ For integer $0\\leq k \\leq {\\rm min}\\{n,q\\}$, let $N_k(u(x),\\ell)$ denote the total number of $v_\\ell(x)$ such that $u(x)+v_\\ell(x)$ has exactly $k$ distinct roots in $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02327","kind":"arxiv","version":1},"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":"1702.02327","created_at":"2026-05-18T00:51:06.211379+00:00"},{"alias_kind":"arxiv_version","alias_value":"1702.02327v1","created_at":"2026-05-18T00:51:06.211379+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.02327","created_at":"2026-05-18T00:51:06.211379+00:00"},{"alias_kind":"pith_short_12","alias_value":"DFPWATHC3WJS","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_16","alias_value":"DFPWATHC3WJS2D44","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_8","alias_value":"DFPWATHC","created_at":"2026-05-18T12:31:10.602751+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/DFPWATHC3WJS2D44GMJCNQUC5T","json":"https://pith.science/pith/DFPWATHC3WJS2D44GMJCNQUC5T.json","graph_json":"https://pith.science/api/pith-number/DFPWATHC3WJS2D44GMJCNQUC5T/graph.json","events_json":"https://pith.science/api/pith-number/DFPWATHC3WJS2D44GMJCNQUC5T/events.json","paper":"https://pith.science/paper/DFPWATHC"},"agent_actions":{"view_html":"https://pith.science/pith/DFPWATHC3WJS2D44GMJCNQUC5T","download_json":"https://pith.science/pith/DFPWATHC3WJS2D44GMJCNQUC5T.json","view_paper":"https://pith.science/paper/DFPWATHC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1702.02327&json=true","fetch_graph":"https://pith.science/api/pith-number/DFPWATHC3WJS2D44GMJCNQUC5T/graph.json","fetch_events":"https://pith.science/api/pith-number/DFPWATHC3WJS2D44GMJCNQUC5T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DFPWATHC3WJS2D44GMJCNQUC5T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DFPWATHC3WJS2D44GMJCNQUC5T/action/storage_attestation","attest_author":"https://pith.science/pith/DFPWATHC3WJS2D44GMJCNQUC5T/action/author_attestation","sign_citation":"https://pith.science/pith/DFPWATHC3WJS2D44GMJCNQUC5T/action/citation_signature","submit_replication":"https://pith.science/pith/DFPWATHC3WJS2D44GMJCNQUC5T/action/replication_record"}},"created_at":"2026-05-18T00:51:06.211379+00:00","updated_at":"2026-05-18T00:51:06.211379+00:00"}