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40 CHAPTER 2. STABLE IDEALSAs a corollary of Theorem 2.17 and Theorem 2.27 we obtain the following algorithm tocompute the Hilbert polynomial of a stable ideal I.Algorithm 2.29. (Computation of the Hilbert polynomial of a stable ideal) Let I ⊂ R bea stable ideal and I g = {m 1 , . . . , m r } a set of minimal generators of I, d i := deg m i thedegree and l i the index of the last variable of the monomial m i , 1 ≤ i ≤ r.The algorithm proceeds in three steps:Input. I g = {m 1 , . . . , m r }, i.e. the set of minimal generators of a stable ideal.1. Compute the Hilbert series H R/I (t) of R/I, i.e.H R/I (t) = 1 − ∑ ri=1 (1 − t)li · t d i(1 − t) n+1 .2. Compute the reduced Hilbert series ˜H R/I (t), i.e.˜H R/I (t) =∑ ki=0 C i · t i(1 − t) d ,where the numerator does not vanish for t = 1.3. Compute the Hilbert polynomial of R/I by using the preceding theorem, i.e.p R/I (z) =k∑( )z + d − 1 − jC j ·.d − 1j=0Output. The Hilbert polynomial p R/I (z) of R/I.Proof. (Correctness) The correctness of the algorithm follows from Theorem 2.17 andTheorem 2.27.We will now use the algorithm in a concrete example. Recall that in Example 2.26 (iii)we did not prove that the computed ideal has the stated Hilbert polynomial. We now givethe proof by using the new algorithm to compute its Hilbert polynomial.Example 2.30. Let R := K[x 0 , x 1 , x 2 , x 3 , x 4 ], and let I be the unique lexicographic ideal,which we obtained in Example 2.26 (iii), i.e. I = (x 0 , x 5 1, x 4 1x 3 2, x 4 1x 2 2x 6 3). We want to computethe Hilbert polynomial according to the preceding algorithm. Thus, we proceed in the threesteps mentioned above:1. The non-reduced Hilbert series H R/I (t) of R/I is1 − (1 − t) 0 · t − (1 − t) · t 5 − (1 − t) 2 · t 7 − (1 − t) 3 · t 12(1 − t) 5 .
2.8. AN APPLICATION TO GOTZMANN’S REGULARITY THEOREM 412. The reduced Hilbert series ˜H R/I (t) of R/I is1 + t + t 2 + t 3 + t 4 − t 7 − t 12 + t 13(1 − t) 3 .3. In the terminology of Theorem 2.27 we have C 0 = C 1 = C 2 = C 3 = C 4 = C 13 = 1,C 7 = C 12 = −1, C j = 0 for all other j and d = 3. Thus, the Hilbert polynomialp R/I (z) of R/I can be written as∑13( )z + d − 1 − j ∑13( ) z + 2 − jp R/I (z) = C j ·= C j ·= 2z 2 + 2z + 1,d − 12j=0which proves the assertion of Example 2.26 (iii).In the last section of this chapter, we want to use the former results in an application toGotzmann’s Regularity Theorem.j=02.8 Appendix: An application to Gotzmann’s RegularityTheoremIn Theorem 1.18, we stated the characterization of Hilbert functions due to F. S. Macaulay,which helped to prove Theorem 2.21. G. Gotzmann carried Macaulay’s work further. Heproved that if the Hilbert function h R/I for I ⊂ R a saturated ideal achieves Macaulay’sbound stated in 1.18 in some degree d, the Hilbert function will always reach Macaulay’sbound in any degree j ≥ d. Additionally, Gotzmann proved an explicit formula of theHilbert polynomial of R/I by the use of the Macaulay representation of the Hilbert functionh R/I (the result is known as Gotzmann’s Persistence Theorem).Furthermore, he was able to prove the so-called Gotzmann’s Regularity Theorem, whichimplies that one can find an upper bound for the degree of the generators of any idealassociated to a given Hilbert polynomial from a special representation of the Hilbert polynomial.We will prove within this appendix that the upper bound established by Gotzmannis the lowest upper bound possible, i.e. we will show, that we can always find an ideal havingminimal generators of the degree of the upper bound proved by Gotzmann. In fact,we will see that the degree of at least one minimal generator of the ideal L p introduced inTheorem 2.25 will reach this bound.To prove this assertion, we first state the two results established by Macaulay and Gotzmanndue to [4], Chapter 3: The Macaulay-Gotzmann Estimates on the Growth of Ideals,Proposition 3.7, Theorem 3.8, p. 150 :
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