{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:IBPXMRJVGBK223O7UY2UFKNECI","short_pith_number":"pith:IBPXMRJV","schema_version":"1.0","canonical_sha256":"405f7645353055ad6ddfa63542a9a412215c641c9bcb1b259b4dca688b71c2be","source":{"kind":"arxiv","id":"2606.08809","version":1},"attestation_state":"computed","paper":{"title":"A complete characterization of a family of permutation trinomials over $\\mathbb F_{p^2}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Marco Timpanella","submitted_at":"2026-06-07T19:59:52Z","abstract_excerpt":"Let $p>3$ be a prime and let $$f_{\\lambda_1,\\lambda_2}(x)=x^{p^2-p+1}+\\lambda_1x^{p^2}+\\lambda_2x^{2p-1}\\in\\mathbb F_{p^2}[x].$$ We determine all pairs $(\\lambda_1,\\lambda_2)\\in(\\mathbb F_{p^2})^2$ for which $f_{\\lambda_1,\\lambda_2}$ is a permutation polynomial of $\\mathbb F_{p^2}$. The final classification consists of three explicit families. The first one is the binomial case $\\lambda_1=0$. The other two are obtained from the condition $\\lambda_2=c\\lambda_1^3$, with $c\\in \\mathbb F_{p}^{*}$, and are defined by two simple equations involving the norm $\\lambda_1^{p+1}$. The proof is based on t"},"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":"2606.08809","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-07T19:59:52Z","cross_cats_sorted":[],"title_canon_sha256":"502ecd5b490b14a316754f53c3a33f3beb759908119c619a29dc345d44ff9aa5","abstract_canon_sha256":"07c8efaea5be5fb627d3fe28057239a4a213dc250f1a368da5331498d65aa562"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T02:07:40.636415Z","signature_b64":"4ZQWng3fDHntSwDus9hsgbiPTcW/ShfZs7sEQhl80vUBWqVQdPfxEO+gINMgqn/+uyBbFsVvVhzJ7gaPUfFiCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"405f7645353055ad6ddfa63542a9a412215c641c9bcb1b259b4dca688b71c2be","last_reissued_at":"2026-06-09T02:07:40.635435Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T02:07:40.635435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A complete characterization of a family of permutation trinomials over $\\mathbb F_{p^2}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Marco Timpanella","submitted_at":"2026-06-07T19:59:52Z","abstract_excerpt":"Let $p>3$ be a prime and let $$f_{\\lambda_1,\\lambda_2}(x)=x^{p^2-p+1}+\\lambda_1x^{p^2}+\\lambda_2x^{2p-1}\\in\\mathbb F_{p^2}[x].$$ We determine all pairs $(\\lambda_1,\\lambda_2)\\in(\\mathbb F_{p^2})^2$ for which $f_{\\lambda_1,\\lambda_2}$ is a permutation polynomial of $\\mathbb F_{p^2}$. The final classification consists of three explicit families. The first one is the binomial case $\\lambda_1=0$. The other two are obtained from the condition $\\lambda_2=c\\lambda_1^3$, with $c\\in \\mathbb F_{p}^{*}$, and are defined by two simple equations involving the norm $\\lambda_1^{p+1}$. 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