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Polynomials that permute the elements of a field, called {\\it permutation polynomials ($PPs$)}, have been the subject of research for decades. Let ${\\mathcal P}^1(\\mathbb{F}_q)$ denote $\\mathbb{Z}_q \\cup \\{\\infty\\}$. If the rational function, $V(x)/U(x)$, permutes the elements of ${\\mathcal P}^1(\\mathbb{F}_q)$, it is called a {\\em permutation rational function (PRf)}. Let $N_d(q)$ denote the number of PPs of degree $d$ over $\\mathbb{F}_q$, and let $N_{v,u}("},"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":"2003.10072","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-03-23T04:07:07Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"fa2f80f0019d81d25fc0602d036cf8f9c8ee8b52315988def888b16f9f071449","abstract_canon_sha256":"24a965a99b921f54b53e784c67b8595a96b9bce8f130a41d7bffe17dbdc6bdeb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:26:11.600526Z","signature_b64":"L6ZgtzX/Bdqu8cYgYQRix2zOGXzgoDdEiru1A7nRjptuuj/9UckiC5fu0e73c0YZNZFS6DYxnYSYOp8ZObNzBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"efb4a792fb3f0ea0066185493c368748074eb509bcc992a029d3d8fc771e6e2c","last_reissued_at":"2026-07-05T02:26:11.600095Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:26:11.600095Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Improved Lower Bounds for Permutation Arrays Using Permutation Rational Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Brian Malouf, I. 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