| ## C.2 Hilbert function Let M 641#641 be a graded module over 642#642 with respect to weights 643#643. The Hilbert function of 13#13, 644#644, is defined (on the integers) by 645#645 The Hilbert-Poincare series of 13#13 is the power series 646#646 It turns out that 647#647 can be written in two useful ways for weights 648#648: 649#649 where 650#650 and 651#651 are polynomials in 652#652. 650#650 is called the first Hilbert series, and 651#651 the second Hilbert series. If 653#653, and 654#654, then 655#655 656#656 (the Hilbert polynomial) for 657#657. Generalizing this to quasihomogeneous modules we get 658#658 where 650#650 is a polynomial in 652#652. 650#650 is called the first (weighted) Hilbert series of M.

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