Corollary 2

*Let $(P_\lambda)_{\lambda \in L}$ be a family of polynomials in $\mathbf{Z}[X_1, \ldots, X_n]$ and let $Q$ be a polynomial in $\mathbf{Z}[X_1, \ldots, X_n]$ such that, for every system of elements $(x_i)_{1 \leq i \leq n}$ belonging to a finite field and for which $P_\lambda(x_1, \ldots, x_n) = 0$ for all $\lambda$, also $Q(x_1, \ldots, x_n) = 0$. Then, if $a$ is the ideal of $\mathbf{Z}[X_1, \ldots, X_n]$ generated by the $P_\lambda$, there exists an integer $m > 0$ such that $Q^m \in a$. Moreover, for every reduced ring $R$ and every system $(y_i)_{1 \leq i \leq n}$ of elements of $R$ such that $P_\lambda(y_1, \ldots, y_n) = 0$ for all $\lambda$, also $Q(y_1, \ldots, y_n) = 0$.

The second assertion follows from the first since the ideal of $\mathbf{Z}[X_1, \ldots, X_n]$ consisting of the polynomials $P$ such that $P(y_1, \ldots, y_n) = 0$ contains $a$. To show the first assertion, it suffices to note that, for every maximal ideal $m$ of $A = \mathbf{Z}[X_1, \ldots, X_n]$ containing $a$, $A/m$ is a finite field (Corollary 1) and the hypothesis implies that the canonical image of $Q$ in $A/m$ is zero; then $Q$ belongs to the intersection of the maximal ideals of $A$ containing $a$, which is the radical of $a$ (Corollary to Proposition 3).