of a graded module
A polynomial expressing the dimensions of the homogeneous components of the module as a function of for large natural numbers . More exactly, the following theorem, demonstrated in essence by D. Hilbert, is valid. Let be a ring of polynomials over a field , graded so that the are homogeneous elements of the first degree, and let be a graded -module of finite type; then there exists a polynomial with rational coefficients such that, for sufficiently large , . This polynomial is called the Hilbert polynomial.
Of greatest interest is the interpretation of the Hilbert polynomial of a graded ring which is the quotient ring of the ring by a homogeneous ideal ; in such a case the Hilbert polynomial gives the projective invariants of the projective variety defined by the ideal . In particular, the degree of coincides with the dimension of , while is said to be the arithmetic genus of . Hilbert polynomials also serve to express the degree of the imbedding . The Hilbert polynomial of the ring is also the name given to the Hilbert polynomial of the projective variety with respect to the imbedding . If is the invertible sheaf corresponding to this imbedding, then
for sufficiently large .
|||D. Hilbert, "Gesammelte Abhandlungen" , 2 , Springer (1933)|
|||M. Baldassarri, "Algebraic varieties" , Springer (1956)|
|||O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975)|
Hilbert polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert_polynomial&oldid=14449