A p-adic Montel Theorem and Locally Polynomial Functions

We prove a version of Montel's Theorem for the case of continuous functions defined over the field Q(p) of p-adic numbers. In particular, we prove that, if Delta(m+1)(h0)f(x) = 0 for all x is an element of Q(p), and h(0) satisfies vertical bar h(0)vertical bar(p) = p (N0), then, for all x(0) is an element of Q(p), the restriction of f over the set x(0) + p(N0)Z(p) coincides with a polynomial p(x0)(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 Delta(m+1)(h)f(x) = 0 (x is an element of X and h is an element of B-X(r) = {x is an element of X : parallel to x parallel to <= r}), whenever f : X -> Y, X is an ultrametric normed space over a non-Archimedean valued field (K,vertical bar . vertical bar) of characteristic zero, and Y is a Q-vector space. By obvious reasons, we call these functions uniformly locally polynomial.