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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $S = k[x_1, \\dots, x_r]$ be the [[polynomial ring|07GNX8GCYP]] with each $x_i$ of degree $1$.\nLet $M = \\bigoplus_{n \\in \\Z} M_n$ be a [[finitely generated|05QY1VEBYD]] [[graded|8KB4SQGC0C]] $S$-module.\nThe Hilbert function of $M$ is $H_M(n) = \\dim_k M_n$.\nThe *Hilbert polynomial* of $M$ is the unique polynomial $P_M \\in \\Q[t]$ such that $H_M(n) = P_M(n)$ for all sufficiently large integers $n$.\n\n## Remarks\n\nSuch a polynomial exists by the [[Hilbert–Serre theorem|HP56DJECMT]], and it is unique because two polynomials that agree at infinitely many integers are equal.\nExample: for $M = k[x, y, z]/(f)$ with $f$ homogeneous of degree $d$, the exact sequence $0 \\to S(-d) \\to S \\to M \\to 0$ gives $H_M(n) = \\binom{n+2}{2} - \\binom{n-d+2}{2}$ for $n \\ge d$, so $P_M(t) = dt - d(d-3)/2$.\nNon-example: the Hilbert function itself need not be a polynomial on all of $\\N$; for $M = k = S/(x_1, \\dots, x_r)$ one has $H_M(0) = 1$ and $H_M(n) = 0$ for $n \\ge 1$, so $P_M = 0$ but $H_M(0) \\neq P_M(0)$.\nFor $M = S$, the Hilbert polynomial is $\\binom{t + r - 1}{r - 1}$, of degree $r - 1$.\n",
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