University of Paderborn Department of Mathematics Diploma Thesis ...

This thesis is organized in four chapters:Chapter 1.Notations and PrerequisitesIn order to enable not only readers with a good background in commutative algebraand algebraic geometry to follow the train of thought presented here, the first chapterof this thesis sets the stage: Basic terminology will be introduced to the readerand basic definitions will be recalled. Some well known results concerning primarydecomposition of ideals, saturation of ideals and Hilbert polynomials will be quoted.Another important result at the end of the first chapter is the characterization of theHilbert function of a homogeneous K-algebra, which goes back to F. S. Macaulayand will not be proved here, either. We will make use of this result, when we provethe correctness of an algorithm stated in Chapter 2. All results, which are statedand not proved at all, can be accepted without the knowledge of any details of theirproof, since we will not make use of these details later.Chapter 2.Stable idealsIn the second chapter, we define Borel-fixed ideals. Under special conditions, wecharacterize these ideals by a certain property (due to [6]), which leads us to theclass of stable ideals. These ideals will be of main interest in the following chapters.The theory to compute the saturation, the Hilbert series or the Hilbert polynomialof a stable ideal is presented. To compute each of these, we give efficient algorithms.Furthermore we establish a link between Hilbert series and stable ideals, i.e. we willdescribe an algorithm to compute a unique saturated lexicographic ideal to a givenHilbert series of K[x 0 , . . . , x n ]/I for a saturated homogeneous ideal I ⊂ K[x 0 , . . . , x n ].We will see that this ideal is also stable. Similarly, we describe how to compute aunique saturated lexicographic ideal to a given Hilbert polynomial, which is again astable ideal. As an appendix of Chapter 2, we deal with an application of the formerresults to a consequence of Gotzmann’s Regularity Theorem (see [5], Chapter 3 ). Inparticular, we point out that Gotzmann’s upper bound on the degrees of the minimalgenerators of ideals to a given Hilbert polynomial cannot be improved.Chapter 3.Operations on stable idealsChapter three is the theoretical core of this thesis. For stable ideals, we define leftshiftsand right-shifts of monomials. We will explain, what has to be understoodunder expansions and contractions of monomials. The reader might think of theseconstructions as algorithmic tools, which will be used to compute all saturated stableideals to a given Hilbert polynomial. Later within this chapter, we will be able to4