University of Paderborn Department of Mathematics Diploma Thesis ...

AbstractThe main result of this thesis is to present an algorithm for the computation of all saturatedstable ideals to a given Hilbert polynomial. It uses and adapts ideas of the Ph.D. thesisCombinatorial Structure on the Hilbert Scheme by Alyson Reeves (see [16]). The class ofstable ideals is introduced, and some useful properties of these ideals are discussed in detailconcerning computation of Hilbert series, Hilbert polynomials and saturation. For each ofthese explicit formulas are given. Special lexicographic ideals are defined, which can beassociated to a given Hilbert series and a given Hilbert polynomial. These ideals play amajor role in the computation of all saturated stable ideals to a given Hilbert polynomial.Apart from the theoretical results of this thesis, the last chapter presents the source codeof an implementation of the algorithm to compute all saturated stable ideals to a givenHilbert polynomial. Additionally, the source code of various other algorithms concerningstable ideals, e.g. to compute Hilbert polynomials, Hilbert series or Hilbert functions isdiscussed in Chapter 4. As a basis for the implementations, the computer algebra systemMuPAD is used. The algorithms can be run on any available version of MuPAD and donot make use of high-level library functions. Therefore the code can easily be implementedin any other comparable computer algebra system.