Let $f \in \mathbb{C}[x_1,...,x_n]$, and let d be the largest $x_i$-degree of f for $0 \leq i \leq n$. Prove that f is the zero polynomial, if $f(a_1,...,a_n)=0$ for all points $(a_1,...,a_n) \in A^n_{\mathbb{C}}$ with $1\leq a_1 \leq d+1$ for all $1 \leq i \leq n$.
I know how to prove that f is the 0 polynomial if it is 0 for all point in $\mathbb{Z}^n$. But here, we cannot use that it is an infinite field. So how would one go about proving this?
Prove it by induction! It's easy in one variable. Then take $f(a_1,\ldots,a_{n-1},x_n)$: it is zero since it has $d+1$ zeros, so all the coefficients must be...