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We determine conditions on the pairs $(A,B)$ and we give lower bounds on the number of pairs $(A,B)$ for which these polynomials permute $\\mathbb{F}_{q^3}$."},"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":"1804.01305","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-04T08:59:12Z","cross_cats_sorted":[],"title_canon_sha256":"1077d8e57aee41289153dc088e0fd8b1cad7f5018f9a0ec7fedb3a54e1c501f9","abstract_canon_sha256":"295009384a8da650f3bdc65d08feab648ad7dff82d8d64d33f4744cf72ccf25f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:15.462814Z","signature_b64":"3BkuGOBzozm+tDEygvVWlVDjOwN1n3ETvJ/oiJ2ucXFpuqrHAqlONGrjx7qyHZJDJqrRGAhrZcgzzx/L4X3EAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ac57e654d752a3bf5d86e111870bfdc7b2ce3cc6e7dbcfa09221094a312a20b","last_reissued_at":"2026-05-18T00:19:15.462075Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:15.462075Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Permutation trinomials over $\\mathbb{F}_{q^3}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniele Bartoli","submitted_at":"2018-04-04T08:59:12Z","abstract_excerpt":"We consider four classes of polynomials over the fields $\\mathbb{F}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+Ax^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+Ax^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+Ax^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+Ax^{q}-Bx$, where $A,B \\in \\mathbb{F}_q$. We determine conditions on the pairs $(A,B)$ and we give lower bounds on the number of pairs $(A,B)$ for which these polynomials permute $\\mathbb{F}_{q^3}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.01305","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":"1804.01305","created_at":"2026-05-18T00:19:15.462204+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.01305v1","created_at":"2026-05-18T00:19:15.462204+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.01305","created_at":"2026-05-18T00:19:15.462204+00:00"},{"alias_kind":"pith_short_12","alias_value":"PLCX4ZKNOUVD","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_16","alias_value":"PLCX4ZKNOUVDX5OY","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_8","alias_value":"PLCX4ZKN","created_at":"2026-05-18T12:32:46.962924+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/PLCX4ZKNOUVDX5OYNYIRQ4F73R","json":"https://pith.science/pith/PLCX4ZKNOUVDX5OYNYIRQ4F73R.json","graph_json":"https://pith.science/api/pith-number/PLCX4ZKNOUVDX5OYNYIRQ4F73R/graph.json","events_json":"https://pith.science/api/pith-number/PLCX4ZKNOUVDX5OYNYIRQ4F73R/events.json","paper":"https://pith.science/paper/PLCX4ZKN"},"agent_actions":{"view_html":"https://pith.science/pith/PLCX4ZKNOUVDX5OYNYIRQ4F73R","download_json":"https://pith.science/pith/PLCX4ZKNOUVDX5OYNYIRQ4F73R.json","view_paper":"https://pith.science/paper/PLCX4ZKN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.01305&json=true","fetch_graph":"https://pith.science/api/pith-number/PLCX4ZKNOUVDX5OYNYIRQ4F73R/graph.json","fetch_events":"https://pith.science/api/pith-number/PLCX4ZKNOUVDX5OYNYIRQ4F73R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PLCX4ZKNOUVDX5OYNYIRQ4F73R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PLCX4ZKNOUVDX5OYNYIRQ4F73R/action/storage_attestation","attest_author":"https://pith.science/pith/PLCX4ZKNOUVDX5OYNYIRQ4F73R/action/author_attestation","sign_citation":"https://pith.science/pith/PLCX4ZKNOUVDX5OYNYIRQ4F73R/action/citation_signature","submit_replication":"https://pith.science/pith/PLCX4ZKNOUVDX5OYNYIRQ4F73R/action/replication_record"}},"created_at":"2026-05-18T00:19:15.462204+00:00","updated_at":"2026-05-18T00:19:15.462204+00:00"}