与Jacobi符号有关的极值序列
Extremal sequences related to the Jacobi symbol
摘要：
对于权重集$A\subseteq \mathbb Z_n$，$A$加权零和常数$C_A(n)$被定义为最小自然数$k$，使得$\mathbb Z_n$中的$k$元素的任何序列具有连续项的$A$加权零和子序列。$\mathbb Z_n$中长度为$C_A(n)-1$的序列，如果不具有任何连续项的$A$加权零和子序列，则称为$A$的$C$极值序列。令$\big(\frac{x}{n}\big)$表示$x\in\mathbb Z_n$的雅可比符号。我们刻画了权集$S(n)=\big\{\,x\in U(n):\big(\frac{x}{n}\big)=1\,\big\}$和权集$L(n;p)=\big\{\,x\in U(n):\big(\frac{x}{n}\big)=\big(\frac{x}{p}\big)\,\big\}$的$C$ -极值序列，其中$p$是$n$的素因子.我们可以用类似于$C$-极值序列的定义的方式为这些权重集定义$D$-极值序列。我们还描述了这些序列。
英文摘要：
For a weight-set $A\subseteq \mathbb Z_n$, the $A$-weighted zero-sum constant $C_A(n)$ is defined to be the smallest natural number $k$, such that any sequence of $k$ elements in $\mathbb Z_n$ has an $A$-weighted zero-sum subsequence of consecutive terms. A sequence of length $C_A(n)-1$ in $\mathbb Z_n$ which does not have any $A$-weighted zero-sum subsequence of consecutive terms will be called a $C$-extremal sequence for $A$. Let $\big(\frac{x}{n}\big)$ denote the Jacobi symbol of $x\in\mathbb Z_n$. We characterize the $C$-extremal sequences for the weight-set $S(n)=\big\{\,x\in U(n):\big(\frac{x}{n}\big)=1\,\big\}$ and for the weight-set $L(n;p)=\big\{\,x\in U(n):\big(\frac{x}{n}\big)=\big(\frac{x}{p}\big)\,\big\}$ where $p$ is a prime divisor of $n$. We can define $D$-extremal sequences for these weight-sets in a way analogous to the definition of $C$-extremal sequences. We also characterize these sequences.
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