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Archivio digitale delle tesi discusse presso l’Università di Pisa

Tesi etd-10012014-102635


Tipo di tesi
Tesi di laurea magistrale
Autore
MASCIA, CARLA
URN
etd-10012014-102635
Titolo
On the Hilbert quasi-polynomials of non-standard graded polynomial rings
Dipartimento
MATEMATICA
Corso di studi
MATEMATICA
Relatori
relatore Dott. Caboara, Massimo
Parole chiave
  • Hilbert function
  • Hilbert quasi-polynomials
  • Hilbert-Poincaré series
  • Non-standard graded rings
Data inizio appello
17/10/2014
Consultabilità
Completa
Riassunto
The Hilbert function, its generating function and the Hilbert polynomial of a graded R-module M have been extensively studied since the famous paper of Hilbert: Ueber die Theorie der algebraischen Formen, 1890. In particular the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry, as they are an efficient way for computing the dimension and the degree of an algebraic variety defined by explicit polynomial equations.

The main aim of this thesis is to generalize the well-know theory of the Hilbert polynomial for standard graded rings to the general case of rings graded by any vector in N^k. The Hilbert function of a non-standard graded ring is of quasi-polynomial type. We investigate the structure of Hilbert quasi-polynomials, such as degree, leading coefficient and proprieties of their coefficients, and at the same time we present an algorithm to calculate it.

The Hilbert quasi-polynomials have application in several fields, for example Bruns and Ichim have used Hilbert quasi-polynomials to give a purely algebraic proof of an old combinatorial result due to Ehrhart, McMullen and Stanley.

The computation of Hilbert-Poincaré series has also received a lot of attention. We consider the problem to get only a part of the Hilbert-Poincaré series, without computing it entirely, and we propose a solution by means of an algorithm due to Roune which exploits the concepts of corners and Koszul simplicial complexes, originally introduced by Bayer.

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