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We represent each polynomial $w$ over $\\mathbb{F}_q$ by \\[w=\\sum_{i=0}^k\\frac{s_i}{Q}{\\left(\\frac{P}{Q}\\right)}^i\\] using a rational base $P/Q$ and digits $s_i\\in\\mathbb{F}_q[X]$ satisfying $\\operatorname{deg}{s_i} < \\operatorname{deg}{P}$. Digit expansions of this type are also defined for formal Laurent series over $\\mathbb{F}_q$. We prove uniqueness and automatic properties of these expansions. Although the $\\omega$-language of the"},"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":"1512.07824","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-12-24T14:30:59Z","cross_cats_sorted":[],"title_canon_sha256":"a195bf6225f1ce0b860ce206517cbb0854a3d5e253484458f30210d466eccccc","abstract_canon_sha256":"f55226daad73f6304c46d05cad816a8718d71033b3a2f9d9a75001ac80f2642e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:20.008654Z","signature_b64":"jYQNan2dK9bgsTfu6sn0vjRBchYntHDCTGSfac+3aHychExQr7BQusU2BCVOudUJZ0MYdGu9gaCiR72TLep4Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"31797f8a103a916f69d9446581a64c357211eb27c99c570659fb749ac3a2a338","last_reissued_at":"2026-05-18T01:03:20.008208Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:20.008208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rational digit systems over finite fields and Christol's Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"J\\\"org M. Thuswaldner, Klaus Scheicher, Manuel Joseph C. Loquias, Mohamed Mkaouar","submitted_at":"2015-12-24T14:30:59Z","abstract_excerpt":"Let $P, Q\\in \\mathbb{F}_q[X]\\setminus\\{0\\}$ be two coprime polynomials over the finite field $\\mathbb{F}_q$ with $\\operatorname{deg}{P} > \\operatorname{deg}{Q}$. We represent each polynomial $w$ over $\\mathbb{F}_q$ by \\[w=\\sum_{i=0}^k\\frac{s_i}{Q}{\\left(\\frac{P}{Q}\\right)}^i\\] using a rational base $P/Q$ and digits $s_i\\in\\mathbb{F}_q[X]$ satisfying $\\operatorname{deg}{s_i} < \\operatorname{deg}{P}$. Digit expansions of this type are also defined for formal Laurent series over $\\mathbb{F}_q$. We prove uniqueness and automatic properties of these expansions. 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