Hilbert polynomial

of a graded module $M = \bigoplus_n M_n$

A polynomial expressing the dimensions of the homogeneous components of the module as a function of $n$ for large natural numbers $n$. More exactly, the following theorem, demonstrated in essence by D. Hilbert, is valid. Let $A = K[X_0,\ldots,X_m]$ be a ring of polynomials over a field $K$, graded so that the $X_i$ are homogeneous elements of the first degree, and let $M = \bigoplus_n M_n$ be a graded $A$-module of finite type; the Hilbert function of $M$ is $h_M(n) = \dim_K M_n$ and there exists a polynomial $P_M(t)$ with rational coefficients such that, for sufficiently large $n$, $P_M(n) = h_M(n) = \dim_K M_n$. This polynomial is called the Hilbert polynomial.

Of greatest interest is the interpretation of the Hilbert polynomial of a graded algebra $R$ which is the quotient ring of the ring $A$ by a homogeneous ideal $I$; in such a case the Hilbert polynomial gives the projective invariants of the projective variety $X = \text{Proj}(R) \subset \mathbf{P}^m$ defined by the ideal $I$. In particular, the degree of $P_R(t)$ coincides with the dimension of $X$, while $P_{\mathrm{A}} = (-1)^{\dim X}(P_R(0)-1)$ is said to be the arithmetic genus of $X$. Hilbert polynomials also serve to express the degree of the imbedding $X \subset \mathbf{P}^m$. The Hilbert polynomial of the ring $R$ is also the name given to the Hilbert polynomial of the projective variety $X$ with respect to the imbedding $X \subset \mathbf{P}^m$. If $\mathcal{O}_X(1)$ is the invertible sheaf corresponding to this imbedding, then $$ P_R(n) = \dim_K H^0(X,\mathcal{O}_X(1)^{{\otimes} n}) $$ for sufficiently large $n$.

References

[1]	D. Hilbert, "Gesammelte Abhandlungen" , 2 , Springer (1933)
[2]	M. Baldassarri, "Algebraic varieties" , Springer (1956)
[3]	O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975)