University of Paderborn Department of Mathematics Diploma Thesis ...

16 CHAPTER 1. NOTATIONS AND PREREQUISITES( )kd−1 + 1but> a ′ by induction hypothesis, which is a contradiction.d − 1Since we have proved k d = max{k k ∈ N, ( kd)≤ a }, one can determine kd , k d−1 , . . . , k sone by one.Definition 1.14. Let d, a ∈ N. We calla =( ) ( )kd kd−1+ + . . . +d d − 1( )ks,swhere k d > k d−1 > . . . > k s ≥ s > 0 for s, k s , . . . , k d ∈ N, the d-th Macaulay representationof a.We have a look at an example:( ( 5 6Example 1.15. Let d := 3 and a := 12. Since = 10 ≤ 12 and = 20 > 12, we( 3)( ( 3)5 2 3have to choose k 3 = 5. From 12 − = 2 and = 1 ≤ 2 < = 3, we obtain( ( 3)2)( 2)( ( 5 2 5 2 1k 2 = 2. Finally 12 − − = 1 provides k 1 = 1. Thus, 12 = + + is3)2)3)2)1)the third Macaulay representation of 12.Definition 1.16. For a, d ∈ N we definea 〈d〉 :=( )kd + 1+d + 1( ) ( )kd kd−1where a = + + . . . +d d − 1case a = 0, we set 0 〈d〉 := 0.( )kd−1 + 1+ . . . +d( )ks + 1,s + 1( )ksis the d-th Macaulay representation of a. In thesExample ( ( 1.17. ( In the terminology of the preceding Example 1.15, we obtain: 12 〈3〉 =6 3 2+ + = 17.4)3)2)With this new terminology, we can characterize Hilbert functions, as mentioned above.Theorem 1.18. (Macaulay’s characterization of Hilbert functions) Let h : Z → Z be afunction. Then h = h R/I for a Hilbert function h R/I and some homogeneous ideal I ⊂ R,if and only if, h(j) = 0 for all j < 0, h(0) = 1 and 0 ≤ h(j + 1) ≤ h(j) 〈j〉 for j > 0.The theorem will not be proved within this thesis. For a proof, see [2]. Instead, we willlook at an example.