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  "body": "\n## Summary\n\nThe multiplicity of a Noetherian [[local ring|MZ02BKPHJA]] $(R, \\mathfrak m)$ of dimension $d$ is the positive integer $e(R)$ such that the [[Hilbert–Samuel function|BNQVS83GZV]] $\\ell(R/\\mathfrak m^n)$ grows like $e(R) n^d / d!$.\nIt measures how singular the corresponding point is.\nA regular local ring has multiplicity $1$, and a plane curve singularity has multiplicity equal to the lowest degree of a term in its equation.\n\n## Explanations\n\n### L1 human\n\nAt a smooth point a curve looks like a single line, and at a node it looks like two lines crossing.\nThe multiplicity counts how many sheets come together at the point, in an algebraic way.\nFor the cusp $y^2 = x^3$ the lowest degree term is $y^2$, of degree $2$, and the multiplicity is $2$.\nThe [[definition|B7CV5DZPRK]] makes this precise.\n",
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