Hilbert polynomial

Remarks

Such a polynomial exists by the Hilbert–Serre theorem, and it is unique because two polynomials that agree at infinitely many integers are equal. Example: 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 $P_M(t) = dt - d(d-3)/2$. Non-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 $H_M(0) \neq P_M(0)$. For $M = S$, the Hilbert polynomial is $\binom{t + r - 1}{r - 1}$, of degree $r - 1$.