Arithmetic of Non-Noetherian Domains and Monoids

An integral domain is called atomic if every non-zero non-unit has a decomposition into a product of finitely many irreducible elements (atoms). If, in addition, every such factorization is unique up to the order of the factors and up to associates, then the domain is called factorial. It is well known that every domain satisfying the ascending chain condition for principal ideals is atomic. If a domain is atomic but not factorial the problem of describing quantitatively the extent to which factoriality fails arises. In the theory of non-unique factorizations one investigates the deviation from being factorial by studying certain arithmetical invariants. The set of all different lengths of factorizations of an element is an example of such an invariant. The hitherto best studied quantity in the context of non-unique factorizations is the system of all sets of lengths arising from factorizations of elements of a given domain. For many types of integral domains of arithmetical interest, e.g., orders in algebraic number fields, it is known that these sets contain a pattern which is repeated very often and where there may be some gaps at the beginning and at the end. In particular, the distance of consecutive lengths of factorizations is bounded. In the project we shall investigate factorization properties of integer-valued polynomials over the rational numbers and, more generally, over algebraic number fields. A polynomial in one variable over the rational numbers is called an integer-valued polynomial if it maps the set of integers to the set of integers. The set of all integer-valued polynomials forms an atomic integral domain. Hitherto arithmetical investigations of this and related domains of integer-valued polynomials suggest that they have a rich and interesting multiplicative structure. The main goal of the project is to study this structure in a systematic way, based on numerous ideal-theoretical investigations of rings of integer-valued polynomials performed in the past.

An integral domain is called atomic if every non-zero non-unit has a decomposition into a product of finitely many irreducible elements (atoms). If, in addition, every such factorization is unique up to the order of the factors and up to associates, then the domain is called factorial. It is well known that every domain satisfying the ascending chain condition for principal ideals is atomic. If a domain is atomic but not factorial the problem of describing quantitatively the extent to which factoriality fails arises. In the theory of non-unique factorizations one investigates the deviation from being factorial by studying certain arithmetical invariants. The set of all different lengths of factorizations of an element is an example of such an invariant. The hitherto best studied quantity in the context of non-unique factorizations is the system of all sets of lengths arising from factorizations of elements of a given domain. For many types of integral domains of arithmetical interest, e.g., orders in algebraic number fields, it is known that these sets contain a pattern which is repeated very often and where there may be some gaps at the beginning and at the end. In particular, the distance of consecutive lengths of factorizations is bounded. In the project we shall investigate factorization properties of integer-valued polynomials over the rational numbers and, more generally, over algebraic number fields. A polynomial in one variable over the rational numbers is called an integer-valued polynomial if it maps the set of integers to the set of integers. The set of all integer-valued polynomials forms an atomic integral domain. Hitherto arithmetical investigations of this and related domains of integer-valued polynomials suggest that they have a rich and interesting multiplicative structure. The main goal of the project is to study this structure in a systematic way, based on numerous ideal-theoretical investigations of rings of integer-valued polynomials performed in the past.