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Register a new userMinimal zero-sum sequence of length five over finite cyclic groups of prime power order
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Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(x_1g)cdotldotscdot(x_lg)$ where $gin G$ and $x_1, ldots, x_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(x_1+cdots+x_l)/ord(g)$ over all possible $gin G$ such that $langle g rangle =G$. Recently the second and the third authors determined the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime order where $S=g^2(x_2g)(x_3g)(x_4g)$. In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime power order. It is shown that if $G=langle grangle$ is a cyclic group of prime power order $n=p^mu$ with $p geq 7$ and $mugeq 2$, and $S=(x_1g)(x_2g)(x_2g)(x_3g)(x_4g)$ with $x_1=x_2$ is a minimal zero-sum sequence with $gcd(n,x_1,x_2,x_3,x_4,x_5)=1$, then $ind(S)=2$ if and only if $S=(mg)(mg)(mfrac{n-1}{2}g)(mfrac{n+3}{2}g)(m(n-3)g)$ where $m$ is a positive integer such that $gcd(m,n)=1$.
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Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(n_1g)cdotldotscdot(n_lg)$ where $gin G$ and $n_1, ldots, n_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(n_1+cdots+n_l)/ord(g)$ over all possible $gin G$ such that $langle g rangle =G$. In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 when $G=langle grangle$ is a cyclic group of a prime order and $S$ has the form $S=g^2(n_2g)(n_3g)(n_4g)$. It is shown that if $G=langle grangle$ is a cyclic group of prime order $p geq 31$, then every minimal zero-sum sequence $S$ of the above mentioned form has index 1 except in the case that $S=g^2(frac{p-1}{2}g)(frac{p+3}{2}g)((p-3)g)$.
Let $p > 155$ be a prime and let $G$ be a cyclic group of order $p$. Let $S$ be a minimal zero-sum sequence with elements over $G$, i.e., the sum of elements in $S$ is zero, but no proper nontrivial subsequence of $S$ has sum zero. We call $S$ is unsplittable, if there do not exist $g$ in $S$ and $x,y in G$ such that $g=x+y$ and $Sg^{-1}xy$ is also a minimal zero-sum sequence. In this paper we show that if $S$ is an unsplittable minimal zero-sum sequence of length $|S|= frac{p-1}{2}$, then $S=g^{frac{p-11}{2}}(frac{p+3}{2}g)^4(frac{p-1}{2}g)$ or $g^{frac{p-7}{2}}(frac{p+5}{2}g)^2(frac{p-3}{2}g)$. Furthermore, if $S$ is a minimal zero-sum sequence with $|S| ge frac{p-1}{2}$, then $ind(S) leq 2$.
Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)cdotldotscdot(n_lg)$ where $gin G$ and $n_1, ldots, n_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(n_1+cdots+n_l)/ord(g)$ over all possible $gin G$ such that $langle g rangle =G$. An open problem on the index of length four sequences asks whether or not every minimal zero-sum sequence of length 4 over a finite cyclic group $G$ with $gcd(|G|, 6)=1$ has index 1. In this paper, we show that if $G=langle grangle$ is a cyclic group with order of a product of two prime powers and $gcd(|G|, 6)=1$, then every minimal zero-sum sequence $S$ of the form $S=(g)(n_2g)(n_3g)(n_4g)$ has index 1. In particular, our result confirms that the above problem has an affirmative answer when the order of $G$ is a product of two different prime numbers or a prime power, extending a recent result by the first author, Plyley, Yuan and Zeng.
We study the maximal cross number $mathsf{K}(G)$ of a minimal zero-sum sequence and the maximal cross number $mathsf{k}(G)$ of a zero-sum 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 $mathsf{k}(G)$ conjectured by Krause and Zahlten hold for $G bigoplus C_{p^a} bigoplus C_{p^b}$ 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 $mathsf{K}(G)$ hold for abelian groups of the form $H_p bigoplus C_{q^m}$ (where $H_p$ is any finite abelian $p$-group) and $C_p bigoplus C_q bigoplus C_r$ for any distinct primes $p,q,r$. We also give a structural result on the minimal zero-sum sequences that achieve this value.
In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and three variables are our main ingredients.
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