FWF Grant P21576-N18

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 the 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 (one-dimensional Krull domains are just 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, let A and B be finite, nonempty subsets, and let A+B = {a+b | a lies in A and b in B } denote 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 1950s) states that |A+B| is at least |A+H| + |B+H| - |H|, where H = {h from G | h+A+B=A+B} 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 Erdös-Ginzburg-Ziv Theorem (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.

Summary

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).

Associated Papers

Zero-Sum Problems with Congruence Conditions A. Geroldinger, D. J. Grynkiewicz, and W. A. Schmid	Acta Math. Hungar. 131 (2011), 323-345. pdf
The Catenary Degree of Krull Monoids I A. Geroldinger, D. J. Grynkiewicz and W. A. Schmid	J. Théor. Nombres Bordeaux 23 (2011), 137-169. pdf
On the Arithmetic of Tame Monoids with Applications to Krull Monoids and Mori Domains A. Geroldinger and F. Kainrath	J. Pure and Applied Algebra 214 (2010), 2199-2218. pdf
A Quantitative Aspect of Non-Unique Factorizations: The Narkiewicz Constants W. D. Gao, A. Geroldinger, and Q. Wang	Int. J. Number Theory 7 (2011), 1463-1502. pdf
On the Davenport Constant and on the Structure of Extremal Zero-Sum Free Sequences A. Geroldinger, M. Liebmann, and A. Philipp	Period. Math. Hungar. 64 (2012). pdf
Semigroup-Theoretical Characterizations of Arithmetical Invariants with Applications to Numerical Monoids and Krull Monoids Victor Blanco, Pedro A. García-Sánchez, and A. Geroldinger	Illinois J. Math, to appear pdf
On Weighted Zero-Sum Sequences S. Adhikari, D. J. Grynkiewicz, and Z. W. Sun	Adv. Appl. Math. to appear pdf
Inverse Additive Problems for Minkowski Sumsets I G. A. Freiman, D. J. Grynkiewicz, O. Serra and Y. Stanchescu	Collectanea Mathematica, to appear pdf
Inverse Additive Problems for Minkowski Sumsets II G. A. Freiman, D. J. Grynkiewicz, O. Serra and Y. Stanchescu	J. Convex Analysis to appear pdf
Structure of general ideal semigroups of monoids and domains A. Reinhart	Journal of Commutative Algebra to apear pdf
Arithmetic Progression Weighted Subsequence Sums D. J. Grynkiewicz, A. Philipp, and V. Ponomarenko	Israel Journal of Mathematics to appear pdf
Products of Two Atoms in Krull Monoids and Arithmetical Characterization of Class Groups P. Baginski, A. Geroldinger, D. J. Grynkiewicz, and A. Philipp	Europ. J. Comb., to appear. pdf
On integral domains that are C-monoids A. Reinhart	Houston J. Math to appear pdf
Non-Commutative Krull Monoids: A Divisor Theoretic Approach and A. Geroldinger	Osaka J. Math., to appear pdf
The Monotone Catenary Degree of Krull Monoids A. Geroldinger and P. Yuan	Results in Mathematics, to appear pdf
The Set of Distances in Krull Monoids A. Geroldinger and P. Yuan	Bull. Lond. Math. Soc, 44 (2012), 1203 -- 1208, pdf
Radical factorial monoids and related concepts A. Reinhart	Annales des sciences mathématiques du Québec, to appear pdf
The Large Davenport Constant I: Groups with a Cyclic, Index 2 Subgroup A. Geroldinger and D. J. Grynkiewicz	J. Pure and Applied Algebra, 217 (2013), 863 - 885, pdf
The Large Davenport Constant II: General Upper Bounds D. J. Grynkiewicz	J. Pure and Applied Algebra, to appear, pdf
Local and Global Tamesness in Krull Monoids W. D. Gao, A. Geroldinger, and W. A. Schmid	Submitted, pdf

Associated Books

Structural Additive Theory
D. J. Grynkiewicz

Developments in Mathematics 30, Springer (June 2013).
Springer Link

Austrian Science Fund (FWF) Grant P21576-N18:

Personnel

Project Summary

Non-Unique Factorizations

Zero-Sum Theory

Associated Papers

Associated Books