Let $k$ be an infinite field, let $n\in \mathbb N$, and identify $\mathbb A_k^n$ with $k^n$ equipped with the Zariski topology whose closed subsets are zero loci of families of polynomials in $k[x_1,\dots,x_n]$. For a subset $Y\subset \mathbb A_k^n$, define its vanishing ideal by
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\begin{align*}
I(Y):=\{f\in k[x_1,\dots,x_n]: f(y)=0 \text{ for every } y\in Y\}.
\end{align*}
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Then $Y$ is Zariski dense in $\mathbb A_k^n$ if and only if