Polynomring/K/Homogenes Polynom/Hilbertfunktion/Fakt

Es sei ${}F\in P=K[X_{1},\ldots ,X_{m}]$ ein homogenes Polynom ${}\neq 0$ vom Grad ${}d$ .

Dann ist die Hilbertfunktion von ${}R=K[X_{1},\ldots ,X_{m}]/(F)$ gleich (die zweite Gleichung gilt für ${}n\geq d$ )

${}H_{R}(n)=H_{P}(n)-H_{P}(n-d)={\frac {d}{(m-2)!}}n^{m-2}+{\text{ kleinere Terme}}\,.$

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