A p-adic Montel theorem and locally polynomial functions

TL;DR

This paper establishes p-adic versions of Jacobi's and Montel's theorems, characterizing continuous functions over bp that satisfy certain difference equations as locally polynomial functions, and computes their general solutions.

Contribution

It extends classical difference theorems to the p-adic setting, providing a characterization of locally polynomial functions over bp and solving related functional equations.

Findings

Functions satisfying  ext{Delta}_{h_0}^{m+1}f=0 are locally polynomial on p-adic balls.

The paper provides explicit descriptions of solutions to  ext{Delta}_h^{m+1}f=0 in ultrametric spaces.

It introduces the concept of uniformly locally polynomial functions over non-Archimedean fields.

Abstract

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 \ \ (x\in X \text{and} h\in B_X(r)=\{x\in X:\|x\|\leq r\}), {equation} whenever $f:X\to Y$, $X$ is an ultrametric normed space over a non-Archimedean valued field $(\mathbb{K},|...|)$ of characteristic zero, and $Y$ is a $\mathbb{Q}$-vector space. By obvious reasons, we call these functions uniformly locally polynomial.