Non-unique factorizations and addition theorems

Non-Unique Factorizations. Let R be a noetherian domain. Then every nonzero element of R that is not a unit has a factorization into atoms of R (these are irreducible elements). But in general, there are many such decompositions, which differ not only up to units and the ordering of the factors. The main objective of factorization theory is to describe and classify the various phenomena of non-unique factorizations in terms of the algebraic invariants of R. If a = u_1 u_k is such a factorization of an element into atoms, then k is called the length of the factorization, and we study the set of lengths L (a) of a (that is, the set of all possible factorization lengths for the element a). Since R is noetherian, all sets of lengths are finite. If R is integrally closed, then R is a Krull domain (the one-dimensional Krull domains are just the Dedekind domains). In that case, the class group G of R is a central invariant controlling the factorizations. In particular, R is factorial if and only if R is a Krull domain with trivial class group. If G is finite, then sets of lengths may become arbitrarily large, but they still have a well-defined structure: they are large subsets of (generalized) arithmetical progressions. Addition Theorems. Let G be an abelian group, A, B finite nonempty subsets, and A+B = {a+b | a lies in A and b in B } their sumset. Direct addition theorems study the size of the sumset in terms of |A| and |B|. For instance, Kneser`s Addition Theorem (proved in the 50s of the previous century) states that |A+B| is at least |A+H| + |B+H| - |H|, where H denotes the stabilizer of A+B. Inverse addition theorems (such as the Theorems of Vosper, Kemperman, and Freiman) give information on the structure of A and B under an assumption that |A+B| is small. Zero-Sum Theory. This field has its origin in Combinatorial Number Theory. Its objective are (finite) sequences S = g_1 g_l over an abelian group G (where in S, the repetition of elements is allowed and their order is disregarded). We say S has sum zero if g_1+ +g_l = 0. A typical direct zero-sum problem studies conditions which ensure that given sequences have nontrivial zero-sum subsequences with prescribed properties. Zero-sum theory is closely connected with various branches of combinatorics, graph theory and geometry. The theorem of Erdös-Ginzburg-Ziv (proved in 1961) is considered as one of the starting points of the area. It states that a sequence S over a finite cyclic group with length |S| at least 2|G|-1 has a zero-sum subsequence of length |G|. It was only in 2007 that this result found its final generalization to groups of rank two. The present project lies in the intersection of the above mentioned areas. Apart from polynomial methods and group algebras, addition theorems are a central tool in zero-sum theory. If the noetherian domain R considered above is integrally closed, then it is a Krull domain, and arithmetical questions in R can be translated into zero-sum problems over its class group G. This transfer process gives optimal results when the class group is finite and every class contains prime divisors (all these assumptions are satisfied for rings of integers in algebraic number fields). A goal of the present project is to apply recent progress in inverse addition theorems and in inverse zero-sum problems to study which differences are possible for sets of lengths (recall that these sets of lengths are large subsets of (generalized) arithmetical progressions).

Non-unique Factorizations. In high school, we all learn about the Fundamental Theorem ofArithmetic:Every positive integer is a product of primes in an essentially unique way.Prime numbers are (multiplicatively) irreducible integers (also called atoms), as for example 2, 3, 5, 7, 11, 13, 17, 19, . On the one side, the study of prime numbers is a classical subfield of Pure Mathematics. On the other side, these results are indispensable to applications in Coding Theory and Cryptography.In mathematics, there are many more domains in which - as in the positive integers - the included objects (numbers, elements) can be written as products or sums of atoms (simpler generic building blocks). However, uniqueness gets lost in general. A good understanding of the variety of different representations (in other words, of the non-uniqueness of factorizations) is helpful for a better understanding of the starting objects.Addition Theorems. A classical addition theorem, due to Lagrange, states that every nonnegative integer is the sum of four squares, i.e., N0= Q + Q + Q + Q where Q is the set of squares. In general, addition theorems in abelian groups provide information on the size and the structure of sumsets A + B, in dependence on the summands A and B.Zero-Sum Theory is a subfield of Combinatorial and Additive Number Theory. It studies finite sequences g1,,gl of terms from abelian groups (the repetition of elements is allowed and their order is disregarded). A central topic is to find conditions which guarantee that given sequences have subsequences whose elements sum up to zero. A further main question asks for the structure of minimal zero-sum sequences.The present project is in the overlap of the above mentioned mathematical subfields, and it is inspired by recent progress in both of them. Using addition theorems we study (minimal) zero-sum sequences whose behaviour models the factorization theoretical behaviour of elements in important mathematical domains.