On polynomial congruences

We deal with functions which fulfil the condition ({Delta_h^{n+1} varphi(x)inmathbb{Z}}) for all x, h taken from some linear space V. We derive necessary and sufficient conditions for such a function to be decent in the following sense: there exist functions ({fcolon Vrightarrow mathbb{R}, gcolon V rightarrow mathbb{Z}}) such that ({varphi = f + g}) and ({Delta_h^{n+1}f(x)=0}) for all ({x, hin V}).