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# Hilbert function of one-dimensional rings /

Material type: BookPublisher: Catania : Scuola Superiore di Catania, 2015Description: 47 p. : ill. ; 24 cm.Subject(s): Group algebras | SemigroupsOnline resources: Abstract
Contents:
Preliminaries -- Numerical semigroups -- One-dimensional local rings -- Hilbert function of the tangent cone of a monomial curve -- Analyzing the non-decrease of the Hilbert function -- Other approaches.
Dissertation note: Tesi di diploma di 2° livello per la Classe delle Scienze Sperimentali Diploma di 2° livello Scuola Superiore di Catania, Catania, Italy 2015 A.A. 2014/2015 Abstract: The Hilbert function is one of the most classical objects associated to a graded ring in order to describe the dimension of the ring itself and the dimensions of its graded components. In this thesis we will focus on the Hilbert function of R when R is a one-dimensional local ring. More specifically, we consider a numerical semigroup ring, i.e. a subalgebra of k[[t]] of the form k[[tg1 ; ... ; tgn]] for some field k and coprime integers g1; ... ; gn 2 N. The numerical semigroup associated to R, which is the set of positive linear combinations of g1; ... ; gn, allows to translate and simplify algebraic and geometric properties of the tangent cone of the correspondent monomial curve in numerical terms. In particular, the non-decrease of the Hilbert function can be expressed as the fact that there are more numbers that can be written in a maximal way as a sum of h elements chosen in fg1; ... : ; gng, than numbers that can be written in a maximal way as a sum of h - 1 elements chosen in fg1; ... ; gng. Although the simple formulation, there are many open problems for the non-decrease of the Hilbert function of a semigroup ring. We will move some steps towards the solution of some of them by considering the numerical invariants of the semigroup associated to the Apéry-set with respect to the multiplicity. We will be able to prove the non-decrease when the multiplicity is 8 and 9 (under one extra condition in the second case). More generally, we will give some necessary and sufficient numerical conditions for the non-decrease. The original results presented in the thesis are contained in the paper On the Hilbert function of the tangent cone of a monomial curve, which is a joint work with Marco D'Anna and Vincenzo Micale, and will soon be published in the journal Semigroup forum.
List(s) this item appears in: Tesi di Laurea, Diploma, Dottorato, Master
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Location Call number Copy number Status Date due
Sala B : Armadio Tesi THS_2015 512.2 D5821 (Browse shelf) 1 Available
Sala B : Armadio Tesi THS_2015 512.2 D5821 (Browse shelf) 2 Available

Tesi di diploma di 2° livello per la Classe delle Scienze Sperimentali Diploma di 2° livello Scuola Superiore di Catania, Catania, Italy 2015 A.A. 2014/2015

Includes bibliographical references (p. 42-45).

Preliminaries -- Numerical semigroups -- One-dimensional local rings -- Hilbert function of the tangent cone of a monomial curve -- Analyzing the non-decrease of the Hilbert function -- Other approaches.

Tesi discussa il 30/11/2015.

The Hilbert function is one of the most classical objects associated to a graded ring in order to describe the dimension of the ring itself and the dimensions of its graded components. In this thesis we will focus on the Hilbert function of R when R is a one-dimensional local ring. More specifically, we consider a numerical semigroup ring, i.e. a subalgebra of k[[t]] of the form k[[tg1 ; ... ; tgn]] for some field k and coprime integers g1; ... ; gn 2 N. The numerical semigroup associated to R, which is the set of positive linear combinations of g1; ... ; gn, allows to translate and simplify algebraic and geometric properties of the tangent cone of the correspondent monomial curve in numerical terms. In particular, the non-decrease of the Hilbert function can be expressed as the fact that there are more numbers that can be written in a maximal way as a sum of h elements chosen in fg1; ... : ; gng, than numbers that can be written in a maximal way as a sum of h - 1 elements chosen in fg1; ... ; gng. Although the simple formulation, there are many open problems for the non-decrease of the Hilbert function of a semigroup ring. We will move
some steps towards the solution of some of them by considering the numerical invariants of the semigroup associated to the Apéry-set with respect to the multiplicity. We will be able to prove the non-decrease when the multiplicity is 8 and 9 (under one extra condition in the second case). More generally, we will give some necessary and sufficient numerical conditions for the non-decrease. The original results presented in the thesis are contained in the paper On the Hilbert function of the tangent cone of a monomial curve, which is a joint work with Marco D'Anna and Vincenzo Micale, and will soon be published in the journal Semigroup forum.

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