I have a natural number $m$, and a polynomial $p: \mathbb{Z}^n \rightarrow \mathbb{Z}$, and I need to find a set $(x_1, x_2,...,x_n)$ so that $p(x_1, x_2,...,x_n) \bmod m = 0$.
First: Does every non-constant polynomial have such a set? And more importantly: Does there exist an (efficient) procedure to determine it, in case a set exists?