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A quantitative aspect of non-unique factorizations: the Narkiewicz constants III

Weidong Gao, Jiangtao Peng, Qinghai Zhong (2013)

Acta Arithmetica

Let K be an algebraic number field with non-trivial class group G and K be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let F k ( x ) denote the number of non-zero principal ideals a K with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that F k ( x ) behaves for x → ∞ asymptotically like x ( l o g x ) 1 - 1 / | G | ( l o g l o g x ) k ( G ) . We prove, among other results, that ( C n C n ) = n + n for all integers n₁,n₂ with 1 < n₁|n₂.

Arithmetic of non-principal orders in algebraic number fields

Andreas Philipp (2010)

Actes des rencontres du CIRM

Let R be an order in an algebraic number field. If R is a principal order, then many explicit results on its arithmetic are available. Among others, R is half-factorial if and only if the class group of R has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.

On delta sets and their realizable subsets in Krull monoids with cyclic class groups

Scott T. Chapman, Felix Gotti, Roberto Pelayo (2014)

Colloquium Mathematicae

Let M be a commutative cancellative monoid. The set Δ(M), which consists of all positive integers which are distances between consecutive factorization lengths of elements in M, is a widely studied object in the theory of nonunique factorizations. If M is a Krull monoid with cyclic class group of order n ≥ 3, then it is well-known that Δ(M) ⊆ {1,..., n-2}. Moreover, equality holds for this containment when each class contains a prime divisor from M. In this note, we consider the question of determining...

On the Delta set of a singular arithmetical congruence monoid

Paul Baginski, Scott T. Chapman, George J. Schaeffer (2008)

Journal de Théorie des Nombres de Bordeaux

If a and b are positive integers with a b and a 2 a mod b , then the set M a , b = { x : x a mod b or x = 1 } is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid M with units M × and any x M M × we say that t is a factorization length of x if and only if there exist irreducible elements y 1 , ... , y t of M and x = y 1 y t . Let ( x ) = { t 1 , ... , t j } be the set of all such lengths (where t i &lt; t i + 1 whenever i &lt; j ). The Delta-set of the element x is defined as the set of gaps in ( x ) : Δ ( x ) = { t i + 1 - t i : 1 i &lt; k } and the Delta-set of the monoid M is given by x M M × Δ ( x ) . We consider the Δ ( M ) when M = M a , b is an ACM with...

Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids

Víctor Blanco, Pedro A. García-Sánchez, Alfred Geroldinger (2010)

Actes des rencontres du CIRM

Arithmetical invariants—such as sets of lengths, catenary and tame degrees—describe the non-uniqueness of factorizations in atomic monoids.We study these arithmetical invariants by the monoid of relations and by presentations of the involved monoids. The abstract results will be applied to numerical monoids and to Krull monoids.

The catenary degree of Krull monoids I

Alfred Geroldinger, David J. Grynkiewicz, Wolfgang A. Schmid (2011)

Journal de Théorie des Nombres de Bordeaux

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree c ( H ) of H is the smallest integer N with the following property: for each a H and each two factorizations z , z of a , there exist factorizations z = z 0 , ... , z k = z of a such that, for each i [ 1 , k ] , z i arises from z i - 1 by replacing at most N atoms from z i - 1 by at most N new atoms. Under a very mild condition...

The set of minimal distances in Krull monoids

Alfred Geroldinger, Qinghai Zhong (2016)

Acta Arithmetica

Let H be a Krull monoid with class group G. Then every nonunit a ∈ H can be written as a finite product of atoms, say a = u 1 · . . . · u k . The set (a) of all possible factorization lengths k is called the set of lengths of a. If G is finite, then there is a constant M ∈ ℕ such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference d ∈ Δ*(H), where Δ*(H) denotes the set of minimal distances of H. We show that max Δ*(H) ≤ maxexp(G)-2,(G)-1 and that equality holds if every...

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