Let $K$ be an infinite field, and $P\in K[x_1,\dots,x_n]$ non-zero (i.e. $P=\sum k_\alpha x^\alpha$ with $x^\alpha=x_1^{\alpha_1}\dots x_n^{\alpha_n}$ and one coefficient $k_\alpha\in K$ is non-zero).
Is there $\bar a\in K^n$ such that $P(\bar a)\neq 0$ ?
I can do $n=1$...
Hint: do it by induction. Prove that if $P$ is a polynomial on an infinite integral domain that has infinitely many roots, then $P=0$.