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  "body": "\n## Summary\n\nLet $M$ be a finitely generated module over a Noetherian [[local ring|MZ02BKPHJA]], and let $I$ be an ideal primary to the maximal ideal.\nThe Hilbert–Samuel function of $M$ is $n \\mapsto \\ell(M/I^n M)$, where $\\ell$ is the [[length|CZSCNZH8VY]].\nFor large $n$ it agrees with a polynomial, the Hilbert–Samuel polynomial.\nIf $M \\neq 0$, the degree of this polynomial is the dimension of $M$, and its leading coefficient gives the [[multiplicity|JX20MNDDXT]].\n\n## Explanations\n\n### L1 human\n\nLook at a ring near a point and ask how much information is visible up to order $n$.\nThe Hilbert–Samuel function counts this, as a length, for each $n$.\nFor the plane at the origin, the polynomials up to degree $n - 1$ are spanned by $n(n+1)/2$ monomials, so the function is $n(n+1)/2$.\nIts growth like $n^2$ reveals that the plane has dimension $2$.\nThe [[definition|22J8E7AB8V]] makes this precise.\n",
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