Please use this identifier to cite or link to this item:
Title: Sumsets and monomial projections of veronese varieties
Author: Llenas i Segura, Sixte Oriol
Director/Tutor: Miró-Roig, Rosa M. (Rosa Maria)
Keywords: Teoria de nombres
Treballs de fi de grau
Àlgebra commutativa
Anells commutatius
Number theory
Bachelor's theses
Commutative algebra
Commutative rings
Issue Date: 24-Jan-2022
Abstract: [en] The main purpose of this paper is to study the so-called sumsets problem. This problem is naturally seen from the point of view of Additive Combinatorics, yet we approach it using Algebraic Geometry. This work is divided into three chapters. The first chapter is devoted to Commutative Algebra. We first define basic concepts, such as graded modules or exact sequences, which will be present throughout the whole article, and then we introduce the concept of the Hilbert function of a graded module. The most important result of the chapter is the fact that this function, for sufficiently large integers, is a polynomial, which we prove by means of the Hilbert-Serre theorem and also Hilbert’s syzygy theorem. Knowing the coefficients of this polynomial is, in general, a very difficult problem. In the second chapter, we link the previous one with Algebraic Geometry. We define the Hilbert function of a projective variety and we calculate it in some simple cases. Next, we study three invariants of projective varieties and introduce the Veronese varieties, which are key in this work. The monomial projections of these varieties will be fundamental to solving the sumsets problem. Finally, in the last chapter, we show that the cardinality of the sumsets can be modeled by the Hilbert function of a suitable monomial projection of a Veronese variety, which proves that this cardinality asymptotically becomes a polynomial.
Note: Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Rosa M. Miró-Roig
Appears in Collections:Treballs Finals de Grau (TFG) - Matemàtiques

Files in This Item:
File Description SizeFormat 
tfg_llenas_segura_sixte_oriol.pdfMemòria561.36 kBAdobe PDFView/Open

This item is licensed under a Creative Commons License Creative Commons