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Let $F$ be an algebraically closed field with $V \subseteq F^n$ and $W \subseteq F^n$ algebraic sets (solutions to sets of polynomial equations with coefficients in $F$). Let $f:V \rightarrow W$ be a regular map between $V$ and $W$.

If $P \in F[X_1,..,X_n]$, then is it true that $$P(v)=0 \Rightarrow P(f(v))=0$$ for $v \in V$?

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Consider the regular map from $F$ to $F$, $f:x \mapsto x^2$. Then $\sqrt{2}$ satisfies $X^2=2$ but $f(\sqrt{2})$ doesn't. –  Bob G May 21 '13 at 14:47

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