Exercise 2, § 2

Assume that $A$ is an infinite integral domain; let $\Phi$ be a set of polynomials $\neq 0$ in the ring $A[X_1, X_2, \ldots, X_n]$. Show that if the power of $\Phi$ is strictly less than that of $A$, then there exists a subset $H$ of $A^n$ equipotent with $A$ such that for each $x = (x_i) \in H$ and for each $f \in \Phi$ we have $f(x) \neq 0$.