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The conditions on $r,q,a$ that are necessary and sufficient for $f$ to be a permutation polynomial (PP) of ${\\Bbb F}_{q^2}$ are not known. (Such conditions are known under an additional assumption that $a^{q+1}=1$.) In this paper, we prove the following: (i) If $f$ is a PP of ${\\Bbb F}_{q^2}$, then $\\text{gcd}(r,q+1)>1$ and $(-a)^{(q+1)/\\text{gcd}(r,q+1)}\\ne 1$. (ii) For a fixed $r>2$ and subject to the conditions that $q+1\\equiv 0\\pmod r$ and $a^{q+1}\\ne 1$, there are only fin","authors_text":"Xiang-dong Hou","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-10-01T22:10:54Z","title":"Permutation Polynomials of $\\Bbb F_{q^2}$ of the form $a{\\tt X}+{\\tt X}^{r(q-1)+1}$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00437","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":"sha256:95f26e539dbf88bf8b633b95d20eebc8a7cf191da2c4f261be91de560162e52a","target":"record","created_at":"2026-05-18T01:31:13Z","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":"65c9d2f7aff7ee9bdf3055be845c4a853b60a3772225936d8284428c08bf757c","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-10-01T22:10:54Z","title_canon_sha256":"976369ab0ea5cafeb74ca979a18c409a9657975d1daf9366b161b37efaf25f56"},"schema_version":"1.0","source":{"id":"1510.00437","kind":"arxiv","version":1}},"canonical_sha256":"22ab71094409fb052dfacec48bee6cde39ba106f30ad80331ae11b90731b3d57","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"22ab71094409fb052dfacec48bee6cde39ba106f30ad80331ae11b90731b3d57","first_computed_at":"2026-05-18T01:31:13.084625Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:31:13.084625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"teDgGN0Kr013qLMAikHg/pePIzBdhTjCxr7fRXTF4oMroK2vMNM/1q/qDEws8W8ZgBwJkeSIfBUBxHEpb+8eAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:31:13.085371Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.00437","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:95f26e539dbf88bf8b633b95d20eebc8a7cf191da2c4f261be91de560162e52a","sha256:4902d5375f75ef7c321817c77e379f491afb833a742b8455e6dce1ae1c451bbd"],"state_sha256":"928826e96fc1f45a247a09a3b07673bd6a8c593e56554349ea8e2d619eb70d3b"}