New upper bounds for the Davenport and for the Erdős-Ginzburg-Ziv constants
type: Item
instance of: scholarly article
title: New upper bounds for the Davenport and for the Erdős-Ginzburg-Ziv constants
zbMATH Open document ID: 1323.11011
author: M. N. Chintamani
author: B. K. Moriya
author: Weidong Gao
author: Prabal Paul
author: R. Thangadurai
published in: Archiv der Mathematik
publication date: 2012-03-23T00:00:00Z
review text: Let \((G, +)\) be a finite abelian group. The Davenport constant \(\mathsf{D}(G)\) of \(G\) is the smallest \(l\) such that each sequence \(g_1, \dots, g_l\) of length \(l\) has a non-empty zero-sum subsequence; that is, there is a non-empty subset \(I \subset \{1, \dots, l\}\) such that \(\sum_{i \in I}g_i = 0\). The Erdős-Ginzburg-Ziv constant \(\mathsf{s}(G)\) is defined in the same way except that \(|I|= \exp(G)\) is imposed as additional condition; that is, there is a zero-sum subsequence of length equal to the exponent. Both constant are intensely studied but in general there exact value is unknown. The paper establishes several new and interesting bounds for these constants, mainly using the Inductive Method. Among others it is shown that \(\mathsf{D}((\mathbb{Z}/n\mathbb{Z})^r) \leq r^{\omega(n)}(n-1)+1\) where \(\omega(n)\) denotes the number of different prime divisors of \(n\). For context we recall that \(\mathsf{D}((\mathbb{Z}/n\mathbb{Z})^r) \geq r(n-1)+1\) and equality is known to hold in case \(n\) is a prime-power or \(r \leq 2\); no example is known where equality does not hold. In addition, a connection of the Davenport constant with the problem of factoring integers using the quadratic sieve is discussed.
Mathematics Subject Classification ID: 11B30
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 11Y05
zbMATH DE Number: 6018031
zbMATH Keywords: Davenport constant
zbMATH Keywords: zero-sum problem
zbMATH Keywords: finite abelian group
zbMATH Keywords: factoring integers using the quadratic sieve
reviewed by: Wolfgang A. Schmid
MaRDI profile type: Publication
full work available at URL: s00013-011-0345-z
OpenAlex ID: W2054444434
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DOI: 10.1007/S00013-011-0345-Z
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description: scientific article; zbMATH DE number 6018031