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  "body": "\n## Summary\n\nLet $P$ be a prime ideal and $n \\ge 1$.\nThe $n$-th symbolic power $P^{(n)}$ is the set of ring elements that land in $P^n$ after multiplication by some element outside $P$.\nIt is the smallest [[primary ideal|TMTPWVETQK]] with radical $P$ that contains $P^n$, so it corrects the fact that $P^n$ itself need not be primary.\nSymbolic powers appear in the proof of [[Krull's principal ideal theorem|SJZW2HN1SC]].\nOver an algebraically closed field, they give the polynomials that vanish to order at least $n$ along a variety.\n\n## Explanations\n\n### L1 human\n\nThe ordinary power $P^n$ can contain extra junk located at points other than the generic point of $P$.\nThe symbolic power throws in everything that belongs to $P^n$ \"near\" $P$, and so keeps only the part of $P^n$ at $P$.\nFor a prime generated by some of the variables of a polynomial ring, such as $(x, y)$ in $k[x, y, z]$, nothing changes and $P^{(n)} = P^n$.\nFor the cone $xy = z^2$ and $P = (x, z)$, the element $x$ lies in $P^{(2)}$ but not in $P^2$.\n",
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