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Let $\\mathbb{F}=\\mathbb{F}_q$ be a prime field.\n  [1.] Suppose $f:\\mathbb{F}^n\\rightarrow \\mathbb{F}$ is a degree five polynomial with bias(f)=\\delta. Then f can be written in the form $f= \\sum_{i=1}^{c} G_i H_i + Q$, where $G_i$ and $H_i$s are nonconstant polynomials satisfying $deg(G_i)+deg(H_i)\\leq 5$ and $Q$ is a degree $\\leq 4$ poly"},"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":"1510.05334","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-19T02:34:36Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"43cddb2c59e295f6141a155917f401ad9b7456ff767199e1608ef667683480b0","abstract_canon_sha256":"c510adb6e2929989f0f81bcb6151c9fc9715a2eead647cb415189f70c283b4af"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:29:52.518222Z","signature_b64":"Fp6rjxqyymEjOO6HQO8fuzpZl59VX9MLDLvq5wfX5d6Vh6WzNAdGdMpW3r+DspTM4MBWFD5wTtipX3WZBdO1BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f7b6c0a764e1252f4bd468bcdf0dfc8d5557f8c885adb922529e55c05eda7ef","last_reissued_at":"2026-05-18T01:29:52.517741Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:29:52.517741Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Structure of Quintic Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Pooya Hatami","submitted_at":"2015-10-19T02:34:36Z","abstract_excerpt":"We study the structure of bounded degree polynomials over finite fields. Haramaty and Shpilka [STOC 2010] showed that biased degree three or four polynomials admit a strong structural property. We confirm that this is the case for degree five polynomials also. Let $\\mathbb{F}=\\mathbb{F}_q$ be a prime field.\n  [1.] Suppose $f:\\mathbb{F}^n\\rightarrow \\mathbb{F}$ is a degree five polynomial with bias(f)=\\delta. Then f can be written in the form $f= \\sum_{i=1}^{c} G_i H_i + Q$, where $G_i$ and $H_i$s are nonconstant polynomials satisfying $deg(G_i)+deg(H_i)\\leq 5$ and $Q$ is a degree $\\leq 4$ poly"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.05334","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":"1510.05334","created_at":"2026-05-18T01:29:52.517822+00:00"},{"alias_kind":"arxiv_version","alias_value":"1510.05334v1","created_at":"2026-05-18T01:29:52.517822+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.05334","created_at":"2026-05-18T01:29:52.517822+00:00"},{"alias_kind":"pith_short_12","alias_value":"B55WYCTWJYJF","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_16","alias_value":"B55WYCTWJYJFF5F5","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_8","alias_value":"B55WYCTW","created_at":"2026-05-18T12:29:14.074870+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/B55WYCTWJYJFF5F5I2F434G7ZD","json":"https://pith.science/pith/B55WYCTWJYJFF5F5I2F434G7ZD.json","graph_json":"https://pith.science/api/pith-number/B55WYCTWJYJFF5F5I2F434G7ZD/graph.json","events_json":"https://pith.science/api/pith-number/B55WYCTWJYJFF5F5I2F434G7ZD/events.json","paper":"https://pith.science/paper/B55WYCTW"},"agent_actions":{"view_html":"https://pith.science/pith/B55WYCTWJYJFF5F5I2F434G7ZD","download_json":"https://pith.science/pith/B55WYCTWJYJFF5F5I2F434G7ZD.json","view_paper":"https://pith.science/paper/B55WYCTW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1510.05334&json=true","fetch_graph":"https://pith.science/api/pith-number/B55WYCTWJYJFF5F5I2F434G7ZD/graph.json","fetch_events":"https://pith.science/api/pith-number/B55WYCTWJYJFF5F5I2F434G7ZD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/B55WYCTWJYJFF5F5I2F434G7ZD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/B55WYCTWJYJFF5F5I2F434G7ZD/action/storage_attestation","attest_author":"https://pith.science/pith/B55WYCTWJYJFF5F5I2F434G7ZD/action/author_attestation","sign_citation":"https://pith.science/pith/B55WYCTWJYJFF5F5I2F434G7ZD/action/citation_signature","submit_replication":"https://pith.science/pith/B55WYCTWJYJFF5F5I2F434G7ZD/action/replication_record"}},"created_at":"2026-05-18T01:29:52.517822+00:00","updated_at":"2026-05-18T01:29:52.517822+00:00"}