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In particular, we prove that, if \\[ \\Delta_{h_0}^{m+1}f(x)=0 \\ \\ \\text{for all} x\\in\\mathbb{Q}_p, \\] and $|h_0|_p=p^{-N_0}$ then, for all $x_0\\in \\mathbb{Q}_p$, the restriction of $f$ over the set $x_0+p^{N_0}\\mathbb{Z}_p$ coincides with a polynomial $p_{x_0}(x)=a_0(x_0)+a_1(x_0)x+...+a_m(x_0)x^m$. Motivated by this result, we compute the general solution of the functional equation with restrictions given by {equation} \\Delta_h^{m+1}f(x)=0 \\ "},"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":"1302.4086","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-02-17T16:48:35Z","cross_cats_sorted":[],"title_canon_sha256":"50e86ff72b1bfc4ac226ffa9e594c4b5ceac9b2f7c92662cda5faec23909f3f4","abstract_canon_sha256":"4b536f0e7ce4126d07aadf5ed4b5ec670902c3c9221353369ee94ec25942a36e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:33:24.218406Z","signature_b64":"lEHm7XRnfpgjj/VdcGyrh0qW1iBXRJsyaCYfp4Me9iF9fHRRu86aOUiX8fXdC2/+/vvBmH5wUvb73w07E/u5Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"369fec6a1bab9bbc577a37172b24ff18dee0b5096593a19e175c5336082cb846","last_reissued_at":"2026-05-18T03:33:24.217657Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:33:24.217657Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A p-adic Montel theorem and locally polynomial functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"J. M. Almira, Kh. F. Abu-Helaiel","submitted_at":"2013-02-17T16:48:35Z","abstract_excerpt":"We prove a version of both Jacobi's and Montel's Theorems for the case of continuous functions defined over the field $\\mathbb{Q}_p$ of $p$-adic numbers. In particular, we prove that, if \\[ \\Delta_{h_0}^{m+1}f(x)=0 \\ \\ \\text{for all} x\\in\\mathbb{Q}_p, \\] and $|h_0|_p=p^{-N_0}$ then, for all $x_0\\in \\mathbb{Q}_p$, the restriction of $f$ over the set $x_0+p^{N_0}\\mathbb{Z}_p$ coincides with a polynomial $p_{x_0}(x)=a_0(x_0)+a_1(x_0)x+...+a_m(x_0)x^m$. Motivated by this result, we compute the general solution of the functional equation with restrictions given by {equation} \\Delta_h^{m+1}f(x)=0 \\ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.4086","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1302.4086","created_at":"2026-05-18T03:33:24.217780+00:00"},{"alias_kind":"arxiv_version","alias_value":"1302.4086v1","created_at":"2026-05-18T03:33:24.217780+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.4086","created_at":"2026-05-18T03:33:24.217780+00:00"},{"alias_kind":"pith_short_12","alias_value":"G2P6Y2Q3VON3","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_16","alias_value":"G2P6Y2Q3VON3YV32","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_8","alias_value":"G2P6Y2Q3","created_at":"2026-05-18T12:27:45.050594+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/G2P6Y2Q3VON3YV32G4LSWJH7DD","json":"https://pith.science/pith/G2P6Y2Q3VON3YV32G4LSWJH7DD.json","graph_json":"https://pith.science/api/pith-number/G2P6Y2Q3VON3YV32G4LSWJH7DD/graph.json","events_json":"https://pith.science/api/pith-number/G2P6Y2Q3VON3YV32G4LSWJH7DD/events.json","paper":"https://pith.science/paper/G2P6Y2Q3"},"agent_actions":{"view_html":"https://pith.science/pith/G2P6Y2Q3VON3YV32G4LSWJH7DD","download_json":"https://pith.science/pith/G2P6Y2Q3VON3YV32G4LSWJH7DD.json","view_paper":"https://pith.science/paper/G2P6Y2Q3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1302.4086&json=true","fetch_graph":"https://pith.science/api/pith-number/G2P6Y2Q3VON3YV32G4LSWJH7DD/graph.json","fetch_events":"https://pith.science/api/pith-number/G2P6Y2Q3VON3YV32G4LSWJH7DD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/G2P6Y2Q3VON3YV32G4LSWJH7DD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/G2P6Y2Q3VON3YV32G4LSWJH7DD/action/storage_attestation","attest_author":"https://pith.science/pith/G2P6Y2Q3VON3YV32G4LSWJH7DD/action/author_attestation","sign_citation":"https://pith.science/pith/G2P6Y2Q3VON3YV32G4LSWJH7DD/action/citation_signature","submit_replication":"https://pith.science/pith/G2P6Y2Q3VON3YV32G4LSWJH7DD/action/replication_record"}},"created_at":"2026-05-18T03:33:24.217780+00:00","updated_at":"2026-05-18T03:33:24.217780+00:00"}