{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:YRIXLW2XHYVVVDU2R7N4YIF7SL","short_pith_number":"pith:YRIXLW2X","schema_version":"1.0","canonical_sha256":"c45175db573e2b5a8e9a8fdbcc20bf92d765914c6bb5c3ba003ebb7cc0d680ed","source":{"kind":"arxiv","id":"2605.26263","version":1},"attestation_state":"computed","paper":{"title":"On planar functions over $\\mathbb{F}_{q^3}$","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Adler Marques, Guilherme Tizziotti, Jo\\~ao Paulo Guardieiro, Luciane Quoos","submitted_at":"2026-05-25T18:45:36Z","abstract_excerpt":"Let $\\mathbb{F}_q$ denote the finite field of order $q$. For $q$ odd, we investigate the planarity over $\\mathbb{F}_{q^3}$ of the family\n  $$\n  f_{E,A,B,C,D}(X) := EX^2+ AX^{q+1}+ BX^{q^2+1}+CX^{2q} +DX^{2q^2}\\in \\mathbb{F}_{q}[X].\n  $$\n  Using results from the theory of q-polynomials, we establish conditions under which these polynomials are planar functions. In particular, we provide characterizations for the planarity property and present new families of planar trinomials, quadrinomials, and pentanomials."},"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":"2605.26263","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-25T18:45:36Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"86b4a7b0ba85b34e4fdfe9c18de91c407d8f65b379461cd52030a06648d41bc4","abstract_canon_sha256":"5913eeb25941caa2ebe04c2877bde88167d59613534b14cf983b2e704d9cc6ca"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-27T01:05:08.799759Z","signature_b64":"XAbDLGJ99FFSD674m0RR3OIK+NWgusF+oSg8IG06bNLbUJtFOH5XrGmHUvOioxuZVCLsY0xAgjWZ1WX6WCZsAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c45175db573e2b5a8e9a8fdbcc20bf92d765914c6bb5c3ba003ebb7cc0d680ed","last_reissued_at":"2026-05-27T01:05:08.798846Z","signature_status":"signed_v1","first_computed_at":"2026-05-27T01:05:08.798846Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On planar functions over $\\mathbb{F}_{q^3}$","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Adler Marques, Guilherme Tizziotti, Jo\\~ao Paulo Guardieiro, Luciane Quoos","submitted_at":"2026-05-25T18:45:36Z","abstract_excerpt":"Let $\\mathbb{F}_q$ denote the finite field of order $q$. For $q$ odd, we investigate the planarity over $\\mathbb{F}_{q^3}$ of the family\n  $$\n  f_{E,A,B,C,D}(X) := EX^2+ AX^{q+1}+ BX^{q^2+1}+CX^{2q} +DX^{2q^2}\\in \\mathbb{F}_{q}[X].\n  $$\n  Using results from the theory of q-polynomials, we establish conditions under which these polynomials are planar functions. In particular, we provide characterizations for the planarity property and present new families of planar trinomials, quadrinomials, and pentanomials."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.26263","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.26263/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":"2605.26263","created_at":"2026-05-27T01:05:08.798963+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.26263v1","created_at":"2026-05-27T01:05:08.798963+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.26263","created_at":"2026-05-27T01:05:08.798963+00:00"},{"alias_kind":"pith_short_12","alias_value":"YRIXLW2XHYVV","created_at":"2026-05-27T01:05:08.798963+00:00"},{"alias_kind":"pith_short_16","alias_value":"YRIXLW2XHYVVVDU2","created_at":"2026-05-27T01:05:08.798963+00:00"},{"alias_kind":"pith_short_8","alias_value":"YRIXLW2X","created_at":"2026-05-27T01:05:08.798963+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/YRIXLW2XHYVVVDU2R7N4YIF7SL","json":"https://pith.science/pith/YRIXLW2XHYVVVDU2R7N4YIF7SL.json","graph_json":"https://pith.science/api/pith-number/YRIXLW2XHYVVVDU2R7N4YIF7SL/graph.json","events_json":"https://pith.science/api/pith-number/YRIXLW2XHYVVVDU2R7N4YIF7SL/events.json","paper":"https://pith.science/paper/YRIXLW2X"},"agent_actions":{"view_html":"https://pith.science/pith/YRIXLW2XHYVVVDU2R7N4YIF7SL","download_json":"https://pith.science/pith/YRIXLW2XHYVVVDU2R7N4YIF7SL.json","view_paper":"https://pith.science/paper/YRIXLW2X","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.26263&json=true","fetch_graph":"https://pith.science/api/pith-number/YRIXLW2XHYVVVDU2R7N4YIF7SL/graph.json","fetch_events":"https://pith.science/api/pith-number/YRIXLW2XHYVVVDU2R7N4YIF7SL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YRIXLW2XHYVVVDU2R7N4YIF7SL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YRIXLW2XHYVVVDU2R7N4YIF7SL/action/storage_attestation","attest_author":"https://pith.science/pith/YRIXLW2XHYVVVDU2R7N4YIF7SL/action/author_attestation","sign_citation":"https://pith.science/pith/YRIXLW2XHYVVVDU2R7N4YIF7SL/action/citation_signature","submit_replication":"https://pith.science/pith/YRIXLW2XHYVVVDU2R7N4YIF7SL/action/replication_record"}},"created_at":"2026-05-27T01:05:08.798963+00:00","updated_at":"2026-05-27T01:05:08.798963+00:00"}