scholarly journals The Inverse Problem Associated to the Davenport Constant for $C_2\oplus C_2 \oplus C_{2n}$, and Applications to the Arithmetical Characterization of Class Groups

10.37236/520 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Wolfgang A. Schmid
Keyword(s):  
Inverse Problem ◽  
Absolute Constant ◽  
Class Group ◽  
Finite Abelian Group ◽  
Maximal Length ◽  
Davenport Constant ◽  
Rings Of Algebraic Integers ◽  
Maximal Multiplicity ◽  
Zero Sum

The inverse problem associated to the Davenport constant for some finite abelian group is the problem of determining the structure of all minimal zero-sum sequences of maximal length over this group, and more generally of long minimal zero-sum sequences. Results on the maximal multiplicity of an element in a long minimal zero-sum sequence for groups with large exponent are obtained. For groups of the form $C_2^{r-1}\oplus C_{2n}$ the results are optimal up to an absolute constant. And, the inverse problem, for sequences of maximal length, is solved completely for groups of the form $C_2^2 \oplus C_{2n}$. Some applications of this latter result are presented. In particular, a characterization, via the system of sets of lengths, of the class group of rings of algebraic integers is obtained for certain types of groups, including $C_2^2 \oplus C_{2n}$ and $C_3 \oplus C_{3n}$; and the Davenport constants of groups of the form $C_4^2 \oplus C_{4n}$ and $C_6^2 \oplus C_{6n}$ are determined.

On zero-sum subsequences of prescribed length

2017 ◽  
Vol 14 (01) ◽  
pp. 167-191 ◽  
Author(s):  
Dongchun Han ◽  
Hanbin Zhang
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Finite Abelian Group ◽  
Davenport Constant ◽  
Zero Sum

Let [Formula: see text] be an additive finite abelian group with exponent [Formula: see text]. For any positive integer [Formula: see text], let [Formula: see text] be the smallest positive integer [Formula: see text] such that every sequence [Formula: see text] in [Formula: see text] of length at least [Formula: see text] has a zero-sum subsequence of length [Formula: see text]. Let [Formula: see text] be the Davenport constant of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite abelian [Formula: see text]-group with [Formula: see text] then [Formula: see text] for every [Formula: see text], which confirms a conjecture by Gao et al. recently, where [Formula: see text] is a prime.

An arithmetic characterization of algebraic number fields with a given class group

1983 ◽  
Vol 94 (1) ◽  
pp. 23-28 ◽  
Author(s):  
David E. Rush
Keyword(s):  
Abelian Group ◽  
Algebraic Number ◽  
Class Group ◽  
Finite Abelian Group ◽  
Number Fields ◽  
Unique Factorization ◽  
Class Groups ◽  
Algebraic Number Fields ◽  
Ring Of Integers

Let R be the ring of integers of a number field K with class group G. It is classical that R is a unique factorization domain if and only if G is the trivial group; and the finite abelian group G is generally considered as a measure of the failure of unique factorization in R. The first arithmetic description of rings of integers with non-trivial class groups was given in 1960 by L. Carlitz (1). He proved that G is a group of order ≤ two if and only if any two factorizations of an element of R into irreducible elements have the same number of factors. In ((6), p. 469, problem 32) W. Narkiewicz asked for an arithmetic characterization of algebraic number fields K with class numbers ≠ 1, 2. This problem was solved for certain class groups with orders ≤ 9 in (2), and for the case that G is cyclic or a product of k copies of a group of prime order in (5). In this note we solve Narkiewicz's problem in general by giving arithmetical characterizations of a ring of integers whose class group G is any given finite abelian group.

scholarly journals On the generalized Davenport constant and the Noether number

2013 ◽  
Vol 11 (9) ◽  
Author(s):  
Kálmán Cziszter ◽  
Mátyás Domokos
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Abelian Group ◽  
Davenport Constant ◽  
Polynomial Invariants ◽  
Arbitrary Finite Group ◽  
Zero Sum ◽  
Zero Vector ◽  
Degree Bound

AbstractKnown results on the generalized Davenport constant relating zero-sum sequences over a finite abelian group are extended for the generalized Noether number relating rings of polynomial invariants of an arbitrary finite group. An improved general upper degree bound for polynomial invariants of a non-cyclic finite group that cut out the zero vector is given.

scholarly journals THE CATENARY DEGREE OF KRULL MONOIDS II

2014 ◽  
Vol 98 (3) ◽  
pp. 324-354 ◽  
Author(s):  
ALFRED GEROLDINGER ◽  
QINGHAI ZHONG
Keyword(s):  
Class Group ◽  
Algebraic Function ◽  
Algebraic Number Field ◽  
Class Groups ◽  
Davenport Constant ◽  
Algebraic Function Field ◽  
Krull Monoids ◽  
Krull Monoid ◽  
Catenary Degree

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$\mathsf{c}(H)$of$H$is the smallest integer$N$with the following property: for each$a\in H$and each pair of factorizations$z,z^{\prime }$of$a$, there exist factorizations$z=z_{0},\dots ,z_{k}=z^{\prime }$of$a$such that, for each$i\in [1,k]$,$z_{i}$arises from$z_{i-1}$by replacing at most$N$atoms from$z_{i-1}$by at most$N$new atoms. To exclude trivial cases, suppose that$|G|\geq 3$. Then the catenary degree depends only on the class group$G$and we have$\mathsf{c}(H)\in [3,\mathsf{D}(G)]$, where$\mathsf{D}(G)$denotes the Davenport constant of$G$. The cases when$\mathsf{c}(H)\in \{3,4,\mathsf{D}(G)\}$have been previously characterized (see Theorem A). Based on a characterization of the catenary degree determined in the paper by Geroldingeret al.[‘The catenary degree of Krull monoids I’,J. Théor. Nombres Bordeaux23(2011), 137–169], we determine the class groups satisfying$\mathsf{c}(H)=\mathsf{D}(G)-1$. Apart from the extremal cases mentioned, the precise value of$\mathsf{c}(H)$is known for no further class groups.

scholarly journals The zero-sum constant, the Davenport constant and their analogues

2020 ◽  
pp. 1-14
Author(s):  
Maciej Zakarczemny
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Asymptotic Behaviour ◽  
Finite Abelian Group ◽  
Classical Case ◽  
Independent Interest ◽  
Davenport Constant ◽  
Smooth Numbers ◽  
Abelian Case ◽  
Zero Sum

Let D(G) be the Davenport constant of a finite Abelian group G. For a positive integer m (the case m=1, is the classical case) let Em(G) (or ηm(G)) be the least positive integer t such that every sequence of length t in G contains m disjoint zero-sum sequences, each of length |G| (or of length ≤exp(G), respectively). In this paper, we prove that if G is an Abelian group, then Em(G)=D(G)–1+m|G|, which generalizes Gao’s relation. Moreover, we examine the asymptotic behaviour of the sequences (Em(G))m≥1 and (ηm(G))m≥1. We prove a generalization of Kemnitz’s conjecture. The paper also contains a result of independent interest, which is a stronger version of a result by Ch. Delorme, O. Ordaz, D. Quiroz. At the end, we apply the Davenport constant to smooth numbers and make a natural conjecture in the non-Abelian case.

scholarly journals Inverse results for weighted Harborth constants

2016 ◽  
Vol 12 (07) ◽  
pp. 1845-1861 ◽  
Author(s):  
Luz E. Marchan ◽  
Oscar Ordaz ◽  
Dennys Ramos ◽  
Wolfgang A. Schmid
Keyword(s):  
Inverse Problem ◽  
Abelian Group ◽  
Inverse Problems ◽  
Cyclic Group ◽  
Finite Abelian Group ◽  
Maximal Length ◽  
Direct Sum ◽  
Inverse Results

For a finite abelian group [Formula: see text], the Harborth constant is defined as the smallest integer [Formula: see text] such that each squarefree sequence over [Formula: see text] of length [Formula: see text] has a subsequence of length equal to the exponent of [Formula: see text] whose terms sum to [Formula: see text]. The plus-minus weighted Harborth constant is defined in the same way except that the existence of a plus-minus weighted subsum equaling [Formula: see text] is required, that is, when forming the sum one can choose a sign for each term. The inverse problem associated to these constants is the problem of determining the structure of squarefree sequences of maximal length that do not yet have such a zero-subsum. We solve the inverse problems associated to these constants for certain groups, in particular, for groups that are the direct sum of a cyclic group and a group of order two. Moreover, we obtain some results for the plus-minus weighted Erdős–Ginzburg–Ziv constant.

Representation of zero-sum invariants by sets of zero-sum sequences over a finite abelian group

2021 ◽  
Author(s):  
Weidong Gao ◽  
Siao Hong ◽  
Wanzhen Hui ◽  
Xue Li ◽  
Qiuyu Yin ◽  
...  
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Zero Sum

scholarly journals Factorizations in Bounded Hereditary Noetherian Prime Rings

2018 ◽  
Vol 62 (2) ◽  
pp. 395-442 ◽  
Author(s):  
Daniel Smertnig
Keyword(s):  
Class Group ◽  
Structure Theory ◽  
Projective Modules ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Prime Rings ◽  
Sets Of Lengths ◽  
Irreducible Elements ◽  
Transfer Homomorphism ◽  
Zero Sum

AbstractIf H is a monoid and a = u1 ··· uk ∈ H with atoms (irreducible elements) u1, … , uk, then k is a length of a, the set of lengths of a is denoted by Ⅼ(a), and ℒ(H) = {Ⅼ(a) | a ∈ H} is the system of sets of lengths of H. Let R be a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisors R• can be written as a product of atoms. We show that if R is bounded and every stably free right R-ideal is free, then there exists a transfer homomorphism from R• to the monoid B of zero-sum sequences over a subset Gmax(R) of the ideal class group G(R). This implies that the systems of sets of lengths, together with further arithmetical invariants, of the monoids R• and B coincide. It is well known that commutative Dedekind domains allow transfer homomorphisms to monoids of zero-sum sequences, and the arithmetic of the latter has been the object of much research. Our approach is based on the structure theory of finitely generated projective modules over HNP rings, as established in the recent monograph by Levy and Robson. We complement our results by giving an example of a non-bounded HNP ring in which every stably free right R-ideal is free but which does not allow a transfer homomorphism to a monoid of zero-sum sequences over any subset of its ideal class group.

scholarly journals Phase Transitions on C*-Algebras from Actions of Congruence Monoids on Rings of Algebraic Integers

2020 ◽  
Author(s):  
Chris Bruce
Keyword(s):  
Phase Transitions ◽  
Class Group ◽  
Number Fields ◽  
Algebraic Integers ◽  
Ring Of Integers ◽  
C Algebras ◽  
Rings Of Algebraic Integers ◽  
Beta 2 ◽  
Factor State ◽  
Type Classification

Abstract We compute the KMS (equilibrium) states for the canonical time evolution on C*-algebras from actions of congruence monoids on rings of algebraic integers. We show that for each $\beta \in [1,2]$, there is a unique KMS$_\beta $ state, and we prove that it is a factor state of type III$_1$. There are phase transitions at $\beta =2$ and $\beta =\infty $ involving a quotient of a ray class group. Our computation of KMS and ground states generalizes the results of Cuntz, Deninger, and Laca for the full $ax+b$-semigroup over a ring of integers, and our type classification generalizes a result of Laca and Neshveyev in the case of the rational numbers and a result of Neshveyev in the case of arbitrary number fields.

scholarly journals An inverse problem for the non-self-adjoint matrix Sturm-Liouville operator

2018 ◽  
Vol 50 (1) ◽  
pp. 71-102 ◽  
Author(s):  
Natalia Pavlovna Bondarenko
Keyword(s):  
Inverse Problem ◽  
Liouville Operator ◽  
Finite Interval ◽  
Spectral Characteristics ◽  
Sufficient Conditions ◽  
Adjoint Matrix ◽  
Method Of Spectral Mappings ◽  
Sturm Liouville ◽  
Necessary And Sufficient

The inverse problem of spectral analysis for the non-self-adjoint matrix Sturm-Liouville operator on a finite interval is investigated. We study properties of the spectral characteristics for the considered operator, and provide necessary and sufficient conditions for the solvability of the inverse problem. Our approach is based on the constructive solution of the inverse problem by the method of spectral mappings. The characterization of the spectral data in the self-adjoint case is given as a corollary of the main result.

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