{"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"}