## Thesis etd-10012014-102635 |

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Thesis type

Tesi di laurea magistrale

Author

MASCIA, CARLA

URN

etd-10012014-102635

Thesis title

On the Hilbert quasi-polynomials of non-standard graded polynomial rings

Department

MATEMATICA

Course of study

MATEMATICA

Supervisors

**relatore**Dott. Caboara, Massimo

Keywords

- Hilbert function
- Hilbert quasi-polynomials
- Hilbert-Poincaré series
- Non-standard graded rings

Graduation session start date

17/10/2014

Availability

Full

Summary

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.

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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