Franz Halter-Koch

Non-unique factorizations : algebraic, combinatorial and analytic theory

CONCEPTS IN FACTORIZATION THEORY AND EXAMPLES Atoms and Primes Free Monoids, Factorial Monoids and Factorizations BF-Monoids Systems of Sets of Lengths FF-Monoids The Catenary Degree and the Tame Degree Rings of Integers of Algebraic Number Fields ALGEBRAIC THEORY OF MONOIDS v-Ideals Prime Ideals and Localizations Complete Integral Closures and Krull Monoids Divisor Homomorphisms and Divisor Theories Krull Monoids and Class Groups Defining Systems and v-Noetherian Monoids Finitary Monoids Class Semigroups C-Monoids and Finitely Primary Monoids Integral Domains Congruence Monoids and Orders ARITHMETIC THEORY OF MONOIDS Finitary Monoids Transfer Principles C-Monoids Saturated Submonoids and Krull Monoids Type Monoids Faithfully Saturated Submonoids Integral Domains and Congruence Monoids Factorizations of Powers of an Element THE STRUCTURE OF SETS OF LENGTHS Multidimensional Arithmetical Progressions Almost Arithmetical Multiprogressions An Abstract Structure Theorem for Sets of Lengths Pattern Ideals and Complete s-Ideals in Finitary Monoids Products of Strongly Primary Monoids and their Submonoids C-Monoids Integral Domains and Congruence Monoids Realization Theorems and Further Examples Sets of Lengths of Powers of an Element ADDITIVE GROUP THEORY Sequences over Abelian Groups Addition Theorems Zero-Sumfree Sequences Cyclic Groups Group Algebras and p-Groups Coverings by Cosets and Elementary p-Groups Short Zero-Sum Sequences and the Inductive Method Groups of Rank Two ARITHMETICAL INVARIANTS OF KRULL MONOIDS The Generalized Davenport Constants The Narkiewicz Constants The Elasticity and Its Refinements The Catenary Degree The Tame Degree Sets of Lengths Containing 2 The Set of Distances and Maximal Half-Factorial Sets Minimal Non-Half-Factorial Sets GLOBAL ARITHMETIC OF KRULL MONOIDS Arithmetical Characterizations of Class Groups I Arithmetical Characterizations of Class Groups II The System of Sets of Lengths for Finite Abelian Groups The System of Sets of Lengths for Infinite Abelian Groups Additively Closed Sequences and Restricted Sumsets Factorization of Large Elements ABSTRACT ANALYTIC NUMBER THEORY Dirichlet Series A General Tauberian Theorem Abstract Formations and Zeta Functions Arithmetical Formations I: Zeta Functions Arithmetical Formations II: Asymptotic Results Arithmetical Formations III: Structure Theory Geometrical Formations I: Asymptotic Results Geometrical Formations II: Structure Theory Algebraic Function Fields Obstructed Formations ANALYTIC THEORY OF NON-UNIQUE FACTORIZATIONS Analytic Theory of Types Elements with Prescribed Factorization Properties The Number of Distinct Factorizations Block-Dependent Factorization Properties APPENDIX A: ABELIAN GROUPS APPENDIX B: COMPLEX ANALYSIS APPENDIX C: THEORY OF INTEGRATION APPENDIX D: POLYHEDRAL CONES BIBLIOGRAPHY LIST OF SYMBOLS SUBJECT INDEX

Kronecker Function Rings and Generalized Integral Closures

Abstract We provide an axiomatic concept of Kronecker function rings and apply it to associate a Kronecker function ring to any integral domain D and any ideal system (star operation) on D. We investigate its behavior in algebraic field extensions and its connection with the defining valuation domains.

Inside factorial monoids and integral domains

Abstract We investigate two classes of monoids and integral domains, called inside and outside factorial, whose definitions are closely related in a divisor-theoretic manner to the concept of unique factorization. We prove that a monoid is outside factorial if and only if it is a Krull monoid with torsion class group, and that it is inside factorial if and only if its root-closure is a rational generalized Krull monoid with torsion class group. We determine the structure of Cale bases of inside factorial monoids and characterize inside factorial monoids among weakly Krull monoids. These characterizations carry over to integral domains. Inside factorial orders in algebraic number fields are characterized by several other factorization properties.

On the asymptotic behaviour of lengths of factorizations

Abstract For an integral domain R and a non-zero non-unit a ∈ R we denote by l ∗ (a) the minimal and by l ∗ (a) the maximal length of a factorization of a into irreducible elements. In this paper, the quantities 1 n l ∗ (a n ) and 1 n l ∗ (a n ) are studied for n→∞, in particular for Krull and certain neotherian domains.

THUE EQUATIONS ASSOCIATED WITH ANKENY–BRAUER–CHOWLA NUMBER FIELDS

For a wide class of one-parameter families of Thue equations of arbitrary degree n [ges ]3 all solutions are determined if the parameter is sufficiently large. The result is based on the Lang–Waldschmidt conjecture, on the primitivity of the associated number fields and on an index bound, which does not depend on the coefficients. By applying the theory of Hilbertian fields and results on thin sets, primitivity is proved for almost all choices (in the sense of density) of the parameters.

Arithmetical semigroups defined by congruences

We investigate multiplicative subsemigroups of N defined by congruences and give necessary and sufficient conditions for the existence of a divisor theory and espacially for unique factorization.

An Artin character and representations of primes by binary quadratic forms II

For any squarefree positive m there exists exactly one solvable antipellian equation, which can be used to construct a certain dihedral extension L/Q, cyclic of degree 4 above k=Q(√−m). We calculate the conductor of L/k and the value of the Artin character of L/k on the corresponding congruence ideal classes of order 2 of k. From this, we deduce results for the representations of powers of primes by binary quadratic forms, in the case where the norm of the fundamental unit of Q(√m) is +1.

On the asymptotic behaviour of the number of distinct factorizations into irreducibles

For an integral domainR and a non-zero non-unitaεR we consider the number of distinct factorizations ofan into irreducible elements ofR for largen. Precise results are obtained for Krull domains and certain noetherian domains. In fact, we prove results valid for certain classes of monoids which then apply to the above-mentioned classes of domains.

Multiplicative ideal theory in the context of commutative monoids

It is well known that large parts of multiplicative ideal theory can be derived in the language of commutative monoids. Classical parts of the theory were treated in this context in my monograph “Ideal Systems” (Marcel Dekker, 1998). The main purpose of this article is to outline some recent developments of multiplicative ideal theory (especially the concepts of spectral star operations and semistar operations together with their applications) in a purely multiplicative setting.

Construction of Ideal Systems with Nice Noetherian Properties

Recently, W. Fanggui and R. L. McCasland [F-McC1], [F-McC2] introduced the notion of a w-envelope M w of a non-zero torsion-free module M over an integral domain D as follows: If K is a quotient field of D and V = KM is the vector space generated by M, then M w consists of all x ∈ V such that Jx ⊂ M for some finitely generated ideal J⊲D satisfying J -1 = D. They called an ideal I ⊲ D a w-ideal if I w = I, and they called D a strong Mori domain if D satisfies the ACC on w-ideals. As a main result, they proved that the w-ideals of a strong Mori domain have a primary decomposition and satisfy the Krull Intersection Theorem and the Krull Principal Ideal Theorem.