University of Paderborn Department of Mathematics Diploma Thesis ...

14 CHAPTER 1. NOTATIONS AND PREREQUISITES(i) Let p(z) = 2z+1 be the given Hilbert polynomial. Now we proceed as in Remark 1.11to compute m 0 and m 1 : In the terminology above we get ( r = 1)and( thus m 1 = ) 1!·2 =z + 1 z + 1 − 22. Then we compute the polynomial ˜p(z) := p(z) − += 2.22Hence, it follows m 0 = 0! · 2 = 2. Thus, we write p(z) = 2z + 1 in the form( ) ( ) ( ( )z + 1 z − 1 z z − 2p(z) = − + − .2 2 1)1(ii) Consider the Hilbert polynomial p(z) = 2z 2 + 2z + 1. In the terminology of Remark1.11, we have r = 2. Following ( the above ) description ( ) provides m 2 = 2! · 2 = 4.z + 2 z − 2Hence, we obtain ˜p(z) := p(z) − + = 6z − 3. Next, we compute3 ( ) 3 ( )z + 1 z − 5m 1 = 1! · 6 = 6 and set ˜p(z) := ˜p(z) − + = 12. Finally, it follows2 2m 0 = 12 and we write p(z) as( ) ( ) ( ) ( ) ( ( )z + 2 z − 2 z + 1 z − 5 z z − 12p(z) = − + − + − .3 3 2 2 1)1We will make use of this special representation of Hilbert polynomials and, in particular,of the sequence m 0 , . . . , m r later. Their values will help to compute the set of generatorsof a special lexicographic ideal, which can be associated to a given Hilbert polynomial p(z).In the last part of this chapter we state a result to characterize Hilbert functions accordingto F.S. Macaulay. It will be useful in various proofs later within this thesis, for examplewhen we will deal with special types of lexicographic ideals: By Macaulay’s characterizationit turns out that the growth of their Hilbert function is minimal.1.3 Characterization of Hilbert functionsAs mentioned above, we will deal with the following question:Given a function h : Z → Z, can one decide whether h : Z → Z is the Hilbert function ofR/I for I ⊂ R a homogeneous ideal?As the heading suggests, we will give an answer to this question. The main result of thissection, which answers the question stated above, was developed by F. S. Macaulay in1927 (see [12] and [13]). Our presentation of the theory follows the treatment in the bookCohen-Macaulay rings by W. Bruns and J. Herzog (see [2], Chapter 4.2 ).( iNote in the following that := 0 whenever i < j.j)