Sumsets and monomial projections of veronese varieties

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.