Factorizations of Algebraic Integers, Block Monoids, and Additive Number Theory

Baginski, Paul; Chapman, Scott T.

doi:10.4169/amer.math.monthly.118.10.901

Cited by 30 publications

(21 citation statements)

References 14 publications

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“…Factorizations and their lengths have been studied extensively under the broad umbrella of factorization theory [21,37,55] (see [36] for a thorough introduction). Investigations usually concern sets of lengths (i.e., without repetition), including asymptotic structure theorems [28,32,35,47] as well as specialized results spanning numerous families of rings and semigroups from number theory [7,8,16], algebra [5,6] and elsewhere (see the survey [33] and the references therein). Several combinatorially-flavored invariants have also been studied (e.g., elasticity [3,40], the delta set [22,39], and the catenary degree [31,34]) to obtain more refined comparisons of length sets across different settings [20].…”

Section: Introductionmentioning

confidence: 99%

Factorization length distribution for affine semigroups I: Numerical semigroups with three generators

Garcia

O’Neill

Yih

2019

European Journal of Combinatorics

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For numerical semigroups with a specified list of (not necessarily minimal) generators, we obtain explicit asymptotic expressions, and in some cases quasipolynomial/quasirational representations, for all major factorization length statistics. This involves a variety of tools that are not standard in the subject, such as algebraic combinatorics (Schur polynomials), probability theory (weak convergence of measures, characteristic functions), and harmonic analysis (Fourier transforms of distributions). We provide instructive examples which demonstrate the power and generality of our techniques. We also highlight unexpected consequences in the theory of homogeneous symmetric functions.2010 Mathematics Subject Classification. 20M14, 05E05.

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Section: Introductionmentioning

confidence: 99%

Factorization length distribution for affine semigroups I: Numerical semigroups with three generators

Garcia

O’Neill

Yih

2019

European Journal of Combinatorics

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“…For further background, we refer the reader to several monographs and conference proceedings [1,23,28,16,11]. It is no surprise that this development has been chronicled over the years by a series of Monthly articles (from [38] to [6,5]). While the focus of this interest has been on commutative domains and their semigroups of ideals, such studies range from abstract semigroup theory to the factorization theory of motion polynomials with application in mechanism science [37].…”

Section: Introductionmentioning

confidence: 99%

Sets of Lengths

Geroldinger¹

2016

The American Mathematical Monthly

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Oftentimes the elements of a ring or semigroup H can be written as finite products of irreducible elements, say a = u 1 · . . . · u k = v 1 · . . . · v ℓ , where the number of irreducible factors is distinct. The set L(a) ⊂ N of all possible factorization lengths of a is called the set of lengths of a, and the full system L(H) = {L(a) | a ∈ H} is a well-studied means of describing the non-uniqueness of factorizations of H. We provide a friendly introduction, which is largely self-contained, to what is known about systems of sets of lengths for rings of integers of algebraic number fields and for transfer Krull monoids of finite type as their generalization.

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“…The following notation is adapted from [2,12,14]. Let (G, +) be a finite abelian group written additively.…”

Section: Introductionmentioning

confidence: 99%

The cross number of minimal zero-sum sequences in finite abelian groups

Kim

2015

Journal of Number Theory

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We study the maximal cross number K(G) of a minimal zerosum sequence and the maximal cross number k(G) of a zerosum free sequence over a finite abelian group G, defined by Krause and Zahlten. In the first part of this paper, we extend a previous result by X. He to prove that the value of k(G) conjectured by Krause and Zahlten holds for G C p a C p b when it holds for G, provided that p and the exponent of G are related in a specific sense. In the second part, we describe a new method for proving that the conjectured value of K(G) holds for abelian groups of the form H p C q m (where H p is any finite abelian p-group) and C p C q C r for any distinct primes p, q, r. We also give a structural result on the minimal zero-sum sequences that achieve this value.

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Factorizations of Algebraic Integers, Block Monoids, and Additive Number Theory

Cited by 30 publications

References 14 publications

Factorization length distribution for affine semigroups I: Numerical semigroups with three generators

Factorization length distribution for affine semigroups I: Numerical semigroups with three generators

Sets of Lengths

The cross number of minimal zero-sum sequences in finite abelian groups

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