University of Paderborn Department of Mathematics Diploma Thesis ...

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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
prove that all saturated stable ideals with the same double saturation and the sameHilbert polynomial are linked by a finite sequence of paired contractions and expansionsof monomials. This provides an algorithmic classification of all such ideals.Additionally, we prove that under special conditions, the unique saturated lexicographicideals to a given Hilbert polynomial and a given Hilbert series defined inChapter 2 have the same double saturation. From these results, we will compute allHilbert series associated to a given Hilbert polynomial p(z) of K[x 0 , . . . , x n ]/I, whereI ⊂ K[x 0 , . . . , x n ] is a saturated stable ideal. In the last section of the chapter wepresent a pseudo code version of the algorithm to compute all saturated stable idealsto a given Hilbert polynomial. The above mentioned contractions and expansions ofmonomials will play an important role within this algorithm.Chapter 4.Algorithms for stable idealsThe aim of Chapter 4 is mainly to use the results presented in the preceding threechapters to establish algorithms for stable ideals. These algorithms will not only bestated in pseudo code, as for example in Chapter 2 and Chapter 3, but also in sourcecode. Here, the MuPAD programming language serves as a basis for the source code.The code listed can always be copied and included into any of the usual versions of thecomputer algebra system MuPAD, which are available today. This code uses little ofthe high level library functions of MuPAD. Since the MuPAD programming languagein its simplest form (the only form, in which it will be used within this thesis) is quitesimilar to the original Pascal programming language, it will be easy for anyone tounderstand the code who is used to Pascal or even any other of the programminglanguages of the modern computer algebra systems available today. Hence, it is alsoeasy to implement the algorithms in any other of the above mentioned programmingenvironments.In detail, we will present the source code for algorithms to compute Hilbert seriesof stable ideals, Hilbert polynomials of stable ideals, special lexicographic ideals, expansionsand contractions and many more. Combining all these algorithms, we willsee that we can compute all saturated stable ideals to a given Hilbert polynomial.The chapter ends with an appendix presenting some experimental results and conclusionsconcerning the algorithm to compute all saturated stable ideals with the sameHilbert Hilbert and the structure of the unique lexicographic ideals associated to aHilbert polynomial.To check the implementations in MuPAD, several tests have been written. The testfilecontaining the tests is not stated within the text, because it would overstep the frameworkof this thesis (it consists of more than 1200 lines of code, which are useful, to checkthe correctness of the algorithms in practice, but not of interest within the mathematicalcontext of this thesis). Nevertheless, the tests are available in electronic form on the CDenclosed to this thesis. On the CD one finds the complete implementation of all algorithms5
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[13] F.S. Macaulay, Some properties
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University of Paderborn Department of Mathematics Diploma Thesis ...

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