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Combinatorial Number Theory and Additive Group Theory

Alfred Geroldinger and Imre Z. Ruzsa
Publication Date: 
Number of Pages: 
Advanced Courses in Mathematics CRM Barcelona
[Reviewed by
Fabio Mainardi
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The volume is divided into three parts: the main theme of the first part is that arithmetical questions in Krull monoids, namely related to factorization, can be translated into combinatorial questions on the class group. The most fundamental examples of Krull monoids are the rings of integers of certain number fields and the monoids of zero-sum sequences over an abelian group (the relevant definitions are provided in the first 10 pages of the book). The main objective of factorization theory in such monoids is the analysis of phenomena related to non-unique factorization. It can be proved, for instance, that the monoid of zero-sum sequences over a finite abelian group is factorial if and only if the group has order at most two. The main strategy in the theory, explained in the first chapter, is to find, for a given monoid H, a simpler monoid B, to study the arithmetic in B, and then “translate” the results back to H via what is called a transfer homomorphism. The main construction of this part of the book is a transfer homomorphism from a Krull monoid to the set of zero-sequences of the class group, allowing to translate multiplicative issues related to factorization, into questions pertaining to additive group theory.

The second third of the volume is a course on combinatorial number theory and can be read independently of the first part. Combinatorial number theory revolves in some sense around Goldbach’s conjecture, which serves as a prototype of the kind of problems involved. (Goldbach’s conjecture asserts that any even integer greater than 2 is a sum of two primes, and every odd integer greater than 3 a sum of three primes.) The most studied object in the theory is called a sumset: for two sets A and B in a group, one defines A+B as the set of all the sums a + b. The cardinality of A+B can lie anywhere between max(m,n) and mn. The main goal in the theory is to understand the relation between the size of A+B and the structures of A and B.

The third part of the book, labelled as “thematic seminars” and written by several authors, is a collection of various research articles.

On the whole, the book is quite technical and aimed principally to researchers or PhD students. The prerequisites are a good acquaintance with general commutative algebra, algebraic number theory and elementary group theory.  

Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are mainly Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. At present, he works in a "classe préparatoire" in Geneva. He may be reached at [email protected].