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 On Subsequence Sums of a Zero-sum Free Sequence

10.37236/970 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Fang Sun
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Zero Sum ◽  
Open Question

Let $G$ be a finite abelian group with exponent $m$, and let $S$ be a sequence of elements in $G$. Let $f(S)$ denote the number of elements in $G$ which can be expressed as the sum over a nonempty subsequence of $S$. In this paper, we show that, if $|S|=m$ and $S$ contains no nonempty subsequence with zero sum, then $f(S)\geq 2m-1$. This answers an open question formulated by Gao and Leader. They proved the same result with the restriction $(m,6)=1$.

On the Maximal Cross Number of Unique Factorization Zero-Sum Sequences over a Finite Abelian Group

Integers ◽  
2012 ◽  
Vol 12 (4) ◽  
Author(s):  
Weidong Gao ◽  
Linlin Wang
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Unique Factorization ◽  
The Cross ◽  
Cross Number ◽  
Zero Sum

Abstract.Letdenote the cross number ofWe determine

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.

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.

On some weighted zero-sum constants

2017 ◽  
Vol 13 (02) ◽  
pp. 301-308 ◽  
Author(s):  
Mohan N. Chintamani ◽  
Prabal Paul
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Upper Bound ◽  
Structural Information ◽  
Finite Abelian Group ◽  
Text Length ◽  
Zero Sum

Let [Formula: see text] be a finite abelian group with exponent exp[Formula: see text]. Let [Formula: see text]. The constant [Formula: see text] is defined as the least positive integer [Formula: see text] such that for any given sequence [Formula: see text] of elements of [Formula: see text] with length [Formula: see text] it has a [Formula: see text] length [Formula: see text]-weighted zero-sum subsequence. In this article, we obtain the exact value of [Formula: see text] for [Formula: see text] and an upper bound for the case [Formula: see text], where [Formula: see text] is an odd prime, [Formula: see text] is an odd integer and [Formula: see text]. We also obtain the structural information on the extremal zero-sum free sequences.

scholarly journals Subsums of a Zero-sum Free Subset of an Abelian Group

10.37236/840 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Weidong Gao ◽  
Yuanlin Li ◽  
Jiangtao Peng ◽  
Fang Sun
Keyword(s):  
Abelian Group ◽  
Lower Bound ◽  
Cyclic Group ◽  
Open Problem ◽  
Nonempty Subset ◽  
Finite Abelian Group ◽  
Positive Answer ◽  
Free Subset ◽  
Zero Sum ◽  
G 28

Let $G$ be an additive finite abelian group and $S \subset G$ a subset. Let f$(S)$ denote the number of nonzero group elements which can be expressed as a sum of a nonempty subset of $S$. It is proved that if $|S|=6$ and there are no subsets of $S$ with sum zero, then f$(S)\geq 19$. Obviously, this lower bound is best possible, and thus this result gives a positive answer to an open problem proposed by R.B. Eggleton and P. Erdős in 1972. As a consequence, we prove that any zero-sum free sequence $S$ over a cyclic group $G$ of length $|S| \ge {6|G|+28\over19}$ contains some element with multiplicity at least ${6|S|-|G|+1\over17}$.

scholarly journals On Subsequence Sums of a Zero-sum Free Sequence II

10.37236/841 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Weidong Gao ◽  
Yuanlin Li ◽  
Jiangtao Peng ◽  
Fang Sun
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Zero Sum ◽  
Zero Group

Let $G$ be an additive finite abelian group with exponent $\exp (G) = n$. For a sequence $S$ over $G$, let f$(S)$ denote the number of non-zero group elements which can be expressed as a sum of a nontrivial subsequence of $S$. We show that for every zero-sum free sequence $S$ over $G$ of length $|S| = n+1$ we have f$(S) \ge 3n-1$.

scholarly journals Subsequence Sums of Zero-sum-free Sequences

10.37236/186 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Pingzhi Yuan
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Zero Sum

Let $G$ be a finite abelian group, and let $S$ be a sequence of elements in $G$. Let $f(S)$ denote the number of elements in $G$ which can be expressed as the sum over a nonempty subsequence of $S$. In this paper, we slightly improve some results of Pixton on $f(S)$ and we show that for every zero-sum-free sequences $S$ over $G$ of length $|S|=\exp(G)+2$ satisfying $f(S)\geq 4\exp(G)-1$.

scholarly journals On the Number of Subsequences with a Given Sum in a Finite Abelian Group

10.37236/620 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Gerard Jennhwa Chang ◽  
Sheng-Hua Chen ◽  
Yongke Qu ◽  
Guoqing Wang ◽  
Haiyan Zhang
Keyword(s):  
Abelian Group ◽  
Lower Bound ◽  
Positive Integer ◽  
Finite Abelian Group ◽  
Zero Sum

Suppose $G$ is a finite abelian group and $S$ is a sequence of elements in $G$. For any element $g$ of $G$, let $N_g(S)$ denote the number of subsequences of $S$ with sum $g$. The purpose of this paper is to investigate the lower bound for $N_g(S)$. In particular, we prove that either $N_g(S)=0$ or $N_g(S)\ge2^{|S|-D(G)+1}$, where $D(G)$ is the smallest positive integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a nonempty zero-sum subsequence. We also characterize the structures of the extremal sequences for which the equality holds for some groups.

scholarly journals Unextendible Sequences in Finite Abelian Groups

10.37236/899 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Jujuan Zhuang
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Minimal Length ◽  
Finite Abelian Groups ◽  
Zero Sum

Let $G=C_{n_1}\oplus \ldots \oplus C_{n_r}$ be a finite abelian group with $r=1$ or $1 < n_1|\ldots|n_r$, and let $S=(a_1,\ldots,a_t)$ be a sequence of elements in $G$. We say $S$ is an unextendible sequence if $S$ is a zero-sum free sequence and for any element $g\in G$, the sequence $Sg$ is not zero-sum free any longer. Let $L(G)=\lceil \log_2{n_1}\rceil+\ldots+\lceil \log_2{n_r}\rceil$ and $d^*(G)=\sum_{i=1}^r(n_i-1)$, in this paper we prove, among other results, that the minimal length of an unextendible sequence in $G$ is not bigger than $L(G)$, and for any integer $k$, where $L(G)\leq k \leq d^*(G)$, there exists at least one unextendible sequence of length $k$.

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