{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:GF4X7CQQHKIW62OZIRSYDJSMGV","short_pith_number":"pith:GF4X7CQQ","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"},"canonical_sha256":"31797f8a103a916f69d9446581a64c357211eb27c99c570659fb749ac3a2a338","source":{"kind":"arxiv","id":"1512.07824","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.07824","created_at":"2026-05-18T01:03:20Z"},{"alias_kind":"arxiv_version","alias_value":"1512.07824v2","created_at":"2026-05-18T01:03:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.07824","created_at":"2026-05-18T01:03:20Z"},{"alias_kind":"pith_short_12","alias_value":"GF4X7CQQHKIW","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"GF4X7CQQHKIW62OZ","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"GF4X7CQQ","created_at":"2026-05-18T12:29:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:GF4X7CQQHKIW62OZIRSYDJSMGV","target":"record","payload":{"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"},"canonical_sha256":"31797f8a103a916f69d9446581a64c357211eb27c99c570659fb749ac3a2a338","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"},"source_kind":"arxiv","source_id":"1512.07824","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:03:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SrWxOuG4hZ8Q9ieQVG3oFHI0OHq4QTXuqKuE+9MLYo+oxSiFHV8sBnNVWWCroIvcG5nyBaU+CaQQcOzTAd4/Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T04:24:47.595517Z"},"content_sha256":"15b0e36485d6601106a0921fbe6f1dd769c2f9505976641ae3644eb1958bb6cd","schema_version":"1.0","event_id":"sha256:15b0e36485d6601106a0921fbe6f1dd769c2f9505976641ae3644eb1958bb6cd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:GF4X7CQQHKIW62OZIRSYDJSMGV","target":"graph","payload":{"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. Although the $\\omega$-language of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.07824","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:03:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7aarsXip7ivpdC0ijD6jk9qURRAIHi4ArrGXDe6XmgOPb2bPaJ50fUheI1bqymvUc/FM6iSg6aYhmbtYgxkvBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T04:24:47.595879Z"},"content_sha256":"025875f0f3746951f6ef81c6578f8f4f157bae1d0d8ab5f6abb80cd99140a658","schema_version":"1.0","event_id":"sha256:025875f0f3746951f6ef81c6578f8f4f157bae1d0d8ab5f6abb80cd99140a658"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GF4X7CQQHKIW62OZIRSYDJSMGV/bundle.json","state_url":"https://pith.science/pith/GF4X7CQQHKIW62OZIRSYDJSMGV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GF4X7CQQHKIW62OZIRSYDJSMGV/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-31T04:24:47Z","links":{"resolver":"https://pith.science/pith/GF4X7CQQHKIW62OZIRSYDJSMGV","bundle":"https://pith.science/pith/GF4X7CQQHKIW62OZIRSYDJSMGV/bundle.json","state":"https://pith.science/pith/GF4X7CQQHKIW62OZIRSYDJSMGV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GF4X7CQQHKIW62OZIRSYDJSMGV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:GF4X7CQQHKIW62OZIRSYDJSMGV","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":"f55226daad73f6304c46d05cad816a8718d71033b3a2f9d9a75001ac80f2642e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-12-24T14:30:59Z","title_canon_sha256":"a195bf6225f1ce0b860ce206517cbb0854a3d5e253484458f30210d466eccccc"},"schema_version":"1.0","source":{"id":"1512.07824","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.07824","created_at":"2026-05-18T01:03:20Z"},{"alias_kind":"arxiv_version","alias_value":"1512.07824v2","created_at":"2026-05-18T01:03:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.07824","created_at":"2026-05-18T01:03:20Z"},{"alias_kind":"pith_short_12","alias_value":"GF4X7CQQHKIW","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"GF4X7CQQHKIW62OZ","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"GF4X7CQQ","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:025875f0f3746951f6ef81c6578f8f4f157bae1d0d8ab5f6abb80cd99140a658","target":"graph","created_at":"2026-05-18T01:03:20Z","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 $P, Q\\in \\mathbb{F}_q[X]\\setminus\\{0\\}$ 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. Although the $\\omega$-language of the","authors_text":"J\\\"org M. Thuswaldner, Klaus Scheicher, Manuel Joseph C. Loquias, Mohamed Mkaouar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-12-24T14:30:59Z","title":"Rational digit systems over finite fields and Christol's Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.07824","kind":"arxiv","version":2},"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":"sha256:15b0e36485d6601106a0921fbe6f1dd769c2f9505976641ae3644eb1958bb6cd","target":"record","created_at":"2026-05-18T01:03:20Z","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":"f55226daad73f6304c46d05cad816a8718d71033b3a2f9d9a75001ac80f2642e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-12-24T14:30:59Z","title_canon_sha256":"a195bf6225f1ce0b860ce206517cbb0854a3d5e253484458f30210d466eccccc"},"schema_version":"1.0","source":{"id":"1512.07824","kind":"arxiv","version":2}},"canonical_sha256":"31797f8a103a916f69d9446581a64c357211eb27c99c570659fb749ac3a2a338","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"31797f8a103a916f69d9446581a64c357211eb27c99c570659fb749ac3a2a338","first_computed_at":"2026-05-18T01:03:20.008208Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:03:20.008208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jYQNan2dK9bgsTfu6sn0vjRBchYntHDCTGSfac+3aHychExQr7BQusU2BCVOudUJZ0MYdGu9gaCiR72TLep4Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:03:20.008654Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.07824","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:15b0e36485d6601106a0921fbe6f1dd769c2f9505976641ae3644eb1958bb6cd","sha256:025875f0f3746951f6ef81c6578f8f4f157bae1d0d8ab5f6abb80cd99140a658"],"state_sha256":"12b815c01dfad747361438b9b1d15b61a5948cb0042c147a4d237f4753a0a1d7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hXnwWVvGF7qff9BgjJHxdQZUSZH041/5GnMWv1uBGmOAKZDM2ZjQwaBUDpYPhOrF0XI2qA8LmLj5dAf2oa6jDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T04:24:47.598029Z","bundle_sha256":"292c6e40876316557af3478ec767d27ba3c1874d564a71ed0e74cd8d0e16f5d0"}}