University of Paderborn Department of Mathematics Diploma Thesis ...

118 CHAPTER 4. ALGORITHMS FOR STABLE IDEALS4.7 Computing all Hilbert functions to a given HilbertpolynomialIn this section, we give an algorithm, which uses the non-reduced Hilbert series of R/I forI ⊂ R a saturated homogeneous ideal (not necessarily stable), to compute the values of theHilbert function h R/I (j). Recall that the Hilbert series of R/I is defined to be the formalpower series H R/I (t) := ∑ h R/I (i) · t i . Since the Hilbert series can always be written asi∈Za rational expression of the formg(t) for a polynomial g(t) with g(1) ≠ 0 and integer(1 − t)dcoefficients and some d ∈ N, we can use the equationg(t)(1 − t) d = ∑ i∈Zh R/I (i) · t ito determine the values h R/I (i) for each i ∈ Z. Because of h R/I (i) = 0 for all i < 0, weonly consider h R/I (i) for i ≥ 0.Remark 4.25. Let I ⊂ R be a saturated homogeneous ideal and H R/I (t) =g(t)(1 − t) the dHilbert series of R/I, where g(t) = a 0 + a 1 · t + a 2 · t 2 + . . . + a m · t m , a 0 , a 1 , . . . , a m ∈ Z,m ∈ N 0 , g(1) ≠ 0 and d ∈ N. Then we obtain the values h R/I (i) for all i > 0 from theequationg(t) = (1 − t) d · ∑h R/I (i) · t ias follows: We write (1 − t) d = b 0 + b 1 · t + b 2 · t 2 + . . . + b d · t d with b 0 , b 1 , . . . , b d ∈ Z, b 0 = 1.Since we know that h R/I (0) = 1, we obtain h R/I (i) for i ≥ 1 successively by the equationsh R/I (1) = a 1 − b 1 · h R/I (0)i∈Zh R/I (2) = a 2 − b 1 · h R/I (1) − b 2 · h R/I (0)h R/I (3) = a 3 − b 1 · h R/I (2) − b 2 · h R/I (1) − b 3 · h R/I (0).h R/I (m) = a m − b 1 · h R/I (m − 1) − b 2 · h R/I (m − 2) − . . . − b m · h R/I (0),which provides the first m + 1 values of h R/I . For all k > m we compute h R/I (k) byh R/I (k) = −b 1 · h R/I (k − 1) − b 2 · h R/I (k − 2) − . . . − b m · h R/I (k − m).The algorithm to determine the values of the Hilbert function h R/I described above is correct,but it lacks elegance: By Theorem 1.7, we know that h R/I (i) = p R/I (i) for all i ≫ 0.If we knew the value of some j ∈ N, for which we obtain p R/I (i) = h R/I (i) for all i ≥ j,we could simply determine the value of the Hilbert function h R/I (i), i ≥ j, by computing