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  "body": "\n## Summary\n\nLet $M$ be a finitely generated graded module over a [[polynomial ring|07GNX8GCYP]] over a field.\nIts Hilbert polynomial gives the dimension of the degree $n$ part of $M$ for all large $n$.\nIts existence is the [[Hilbert–Serre theorem|HP56DJECMT]].\nFor the coordinate ring of a projective variety, the degree of the Hilbert polynomial is the dimension of the variety.\nIts leading coefficient encodes the degree of the variety.\n\n## Explanations\n\n### L1 human\n\nCount how many independent homogeneous polynomials of degree $n$ survive on some shape.\nFor large $n$ this count follows a polynomial formula in $n$.\nFor a plane curve given by one equation of degree $d$, the count is $dn - d(d-3)/2$ for large $n$.\nThe growth rate of the count tells the dimension of the shape.\nThe leading coefficient tells its degree, the number of points in which it meets a general linear space of complementary dimension.\nFor instance, a plane curve of degree $d$ meets a general line in $d$ points.\nThe [[definition|5FR6JX04RS]] makes this precise.\n",
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    "name": "Hilbert polynomial",
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  "msc": [
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  "name": "Hilbert polynomial",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/RMH/RMH4FBCFDA.md",
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  "tag": "RMH4FBCFDA",
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  "url": "https://math2.tamnd.com/wiki/hilbert-polynomial/",
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    "tag": "RMH4FBCFDA",
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