A p-adic Montel theorem and locally polynomial functions

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.Comment: 12 pages, submitted to a journa

A note on invariant subspaces and the solution of some classical functional equations

We study the continuous solutions of several classical functional equations by using the properties of the spaces of continuous functions which are invariant under some elementary linear trans-formations. Concretely, we use that the sets of continuous solutions of certain equations are closed vector subspaces of $C(\mathbb{C}^d,\mathbb{C})$ which are invariant under affine transformations $T_{a,b}(f)(z)=f(az+b)$, or closed vector subspaces of $C(\mathbb{R}^d,\mathbb{R})$ which are translation and dilation invariant. These spaces have been recently classified by Sternfeld and Weit, and Pinkus, respectively, so that we use this information to give a direct characterization of the continuous solutions of the corresponding functional equations.Comment: 7 pages, submitted to a Journa

A qualitative description of graphs of discontinuous polynomial functions

We prove that, if f:R^n\to R satisfies Fr\'echet's functional equation and f(x_1,...,x_n) is not an ordinary algebraic polynomial in the variables x_1,...,x_n, then f is unbounded on all non-empty open set U of R^n. Furthermore, the closure of its graph contains an unbounded open set.Comment: 9 pages, submitted to a journa