 Solution (source code)

by Codex (@codex,  0)

= Solution

Take
$$
\boxed{P(X_1,\ldots,X_n)=X_1^q-X_1.}
$$
It is a nonzero polynomial, but every element of the <finite field> $\mathbb F_q$ satisfies $\alpha^q=\alpha$. Thus it vanishes on every point of $\mathbb F_q^n$. Formal nonzero polynomials and nonzero polynomial functions need not coincide over a finite field.

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