| ## C.2 Hilbert function Let M 639#639 be a graded module over 640#640 with respect to weights 641#641. The Hilbert function of 13#13, 642#642, is defined (on the integers) by 643#643 The Hilbert-Poincare series of 13#13 is the power series 644#644 It turns out that 645#645 can be written in two useful ways for weights 646#646: 647#647 where 648#648 and 649#649 are polynomials in 650#650. 648#648 is called the first Hilbert series, and 649#649 the second Hilbert series. If 651#651, and 652#652, then 653#653 654#654 (the Hilbert polynomial) for 655#655. Generalizing this to quasihomogeneous modules we get 656#656 where 648#648 is a polynomial in 650#650. 648#648 is called the first (weighted) Hilbert series of M.

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