Affine Monoids, Hilbert Bases and Hilbert Functions

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Titel: Affine Monoids, Hilbert Bases and Hilbert Functions
Autor(en): Koch, Robert
Erstgutachter: Prof. Dr. Winfried Bruns
Zweitgutachter: Prof. Dr. Udo Vetter
Zusammenfassung: The aim of this thesis is to introduce the reader to the theory of affine monoids and, thereby, to present some results. We therefore start with some auxiliary sections, containing general introductions to convex geometry, affine monoids and their algebras, Hilbert functions and Hilbert series. One central part of the thesis then is the description of an algorithm for computing the integral closure of an affine monoid. The algorithm has been implemented, in the computer program `normaliz´; it outputs the Hilbert basis and the Hilbert function of the integral closure (if the monoid is positive). Possible applications include: finding the lattice points in a lattice polytope, computing the integral closure of a monomial ideal and solving Diophantine systems of linear inequalities. The other main part takes up the notion of multigraded Hilbert function: we investigate the effect of the growth of the Hilbert function along arithmetic progressions (within the grading set) on global growth. This study is motivated by the case of a finitely generated module over a homogeneous ring: there, the Hilbert function grows with a degree which is well determined by the degree of the Hilbert polynomial (and the Krull dimension).
URL: https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2003071115
Schlagworte: affine monoid; integral closure; normalization; Hilbert basis; Hilbert function; affine monoid algebra; semigroup ring
Erscheinungsdatum: 11-Jul-2003
Enthalten in den Sammlungen:FB06 - E-Dissertationen

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