{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:QNE5JSQV7TLMG2FFYUSOZ4G53J","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"de32f7f5b8be8d2fce12347728f68e562319295c71e3193a913740d490742d7b","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-14T18:58:08Z","title_canon_sha256":"2adb81ae0e19a048ff6a3d9b54845f77206b22ba8e7ee87e633524f11e268a38"},"schema_version":"1.0","source":{"id":"1208.2942","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.2942","created_at":"2026-05-18T03:48:46Z"},{"alias_kind":"arxiv_version","alias_value":"1208.2942v1","created_at":"2026-05-18T03:48:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.2942","created_at":"2026-05-18T03:48:46Z"},{"alias_kind":"pith_short_12","alias_value":"QNE5JSQV7TLM","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"QNE5JSQV7TLMG2FF","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"QNE5JSQV","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:45dfe1a98826dc18831b7fa77b89bc90b01ddf5dd75a4bfdd7281dd907057304","target":"graph","created_at":"2026-05-18T03:48:46Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $p$ be a prime and $q$ a power of $p$. For $n\\ge 0$, let $g_{n,q}\\in\\Bbb F_p[{\\tt x}]$ be the polynomial defined by the functional equation $\\sum_{a\\in\\Bbb F_q}({\\tt x}+a)^n=g_{n,q}({\\tt x}^q-{\\tt x})$. When is $g_{n,q}$ a permutation polynomial (PP) of $\\Bbb F_{q^e}$? This turns out to be a challenging question with remarkable breath and depth, as shown in the predecessor of the present paper. We call a triple of positive integers $(n,e;q)$ {\\em desirable} if $g_{n,q}$ is a PP of $\\Bbb F_{q^e}$. In the present paper, we find many new classes of desirable triples whose corresponding PPs we","authors_text":"Neranga Fernando, Stephen D. Lappano, Xiang-dong Hou","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-14T18:58:08Z","title":"A New Approach to Permutation Polynomials over Finite Fields, II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.2942","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:0f113156e244d22867e13a1e65b2367f404caf9aca7a4f770dae29026d034919","target":"record","created_at":"2026-05-18T03:48:46Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"de32f7f5b8be8d2fce12347728f68e562319295c71e3193a913740d490742d7b","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-14T18:58:08Z","title_canon_sha256":"2adb81ae0e19a048ff6a3d9b54845f77206b22ba8e7ee87e633524f11e268a38"},"schema_version":"1.0","source":{"id":"1208.2942","kind":"arxiv","version":1}},"canonical_sha256":"8349d4ca15fcd6c368a5c524ecf0ddda530b33611211e62239e52c5e45988b4c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8349d4ca15fcd6c368a5c524ecf0ddda530b33611211e62239e52c5e45988b4c","first_computed_at":"2026-05-18T03:48:46.856268Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:48:46.856268Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uEkLi7N+R4bJ65BiK2STsOXNnMv4mnYE75L59IpZl2lABwJPkbHH9BmwtrB9JwkBEEcqHIoXJ/qOQy8xoGDZDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:48:46.856803Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.2942","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0f113156e244d22867e13a1e65b2367f404caf9aca7a4f770dae29026d034919","sha256:45dfe1a98826dc18831b7fa77b89bc90b01ddf5dd75a4bfdd7281dd907057304"],"state_sha256":"74db56e110837b35cd12cb5ce7408d2bda53e2feeae082357c0a4321b613c8f4"}