Exercise 1, § 2

Suppose $A$ is an integral domain; let $f$ be a polynomial of the ring $A[X_1, X_2, \ldots, X_n]$ of degree $\leq k_i$ in $X_i$ (for $1 \leq i \leq n$). For each value of $i$ ($1 \leq i \leq n$) let $H_i$ be a set of $k_i + 1$ elements of $A$. Show that if $f(x_1, x_2, \ldots, x_n) = 0$ for each element $(x_i) \in \prod_{i=1}^n H_i$, then $f = 0$.