We prove a version of both Jacobi's and Montel's Theorems for the case of
continuous functions defined over the field Qpβ of p-adic numbers.
In particular, we prove that, if Ξh0βm+1βf(x)=0Β Β forΒ allxβQpβ, and β£h0ββ£pβ=pβN0β then, for all x0ββQpβ, the restriction of f over the set x0β+pN0βZpβ
coincides with a polynomial px0ββ(x)=a0β(x0β)+a1β(x0β)x+...+amβ(x0β)xm.
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:XβY,
X is an ultrametric normed space over a non-Archimedean valued field
(K,β£...β£) of characteristic zero, and Y is a Q-vector
space. By obvious reasons, we call these functions uniformly locally
polynomial.Comment: 12 pages, submitted to a journa