论文标题
零和无元素和超平面布置
Zero-sum-free tuples and hyperplane arrangements
论文作者
论文摘要
向量$（v_ {1}，v_ {2}，\ cdots，v_ {d}）$中的$ \ mathbb {z} _n^{d} $，据说如果没有$ d $ -tuple，则如果没有其组件的$ n属于$ zer $ zer的零n，则是$ d $ -tuple。我们用$α_n^d $表示此系列的基数。我们让$β_n^d $表示$ \ mathbb {z} _n^{d} $中$ \ gcd（v_1，\ cdots，v_d，v_d，n）= 1 $的零和无元组的基数。我们表明，$α_n^d = ϕ（n）\ binom {n-1} {d} $当$ d> n/2 $时，在一般情况下，我们证明了递归公式，划分结果，边界和$α_n_n^d $和$β_n^d $ $α_n_n^d $ $α_n_n^d $的差异结果。特别是，$α_n^{n-1} =β_n^1 = ϕ（n）$，这表明可以将这些序列视为Euler的基本功能的概括。我们还将计算$α_n^d $计算的问题与计数点相关联，在$ \ mathbb {z} _n $上定义的某个超平面布置的补充中。结果表明，超平面布置的特征多项式捕获了所有整数$ n $的$α_n^d $，这些$ n $对于某些决定因素是相对典型的。我们研究数字$α_n^{d} $的行和列模式。我们表明，对于任何固定的$ d $，$ \ {α_n^d \} $渐近等同于$ \ {n^d \} $。我们还显示了$β_n^d $的渐近生长与Riemann Zeta函数$ζ（D）$的价值之间的联系。最后，我们表明$α_n^d $自然出现在有限领域产品中的Mathieu-Zhao子空间的研究中。
A vector $(v_{1}, v_{2}, \cdots, v_{d})$ in $\mathbb{Z}_n^{d}$ is said to be a zero-sum-free $d$-tuple if there is no non-empty subset of its components whose sum is zero in $\mathbb{Z}_n$. We denote the cardinality of this collection by $α_n^d$. We let $β_n^d$ denote the cardinality of the set of zero-sum-free tuples in $\mathbb{Z}_n^{d}$ where $\gcd(v_1, \cdots,v_d, n) = 1$. We show that $α_n^d=ϕ(n)\binom{n-1}{d}$ when $d > n/2$, and in the general case, we prove recursive formulas, divisibility results, bounds, and asymptotic results for $α_n^d$ and $β_n^d$. In particular, $α_n^{n-1} = β_n^1= ϕ(n)$, suggesting that these sequences can be viewed as generalizations of Euler's totient function. We also relate the problem of computing $α_n^d$ to counting points in the complement of a certain hyperplane arrangement defined over $\mathbb{Z}_n$. It is shown that the hyperplane arrangement's characteristic polynomial captures $α_n^d$ for all integers $n$ that are relatively prime to some determinants. We study the row and column patterns in the numbers $α_n^{d}$. We show that for any fixed $d$, $\{α_n^d \}$ is asymptotically equivalent to $\{ n^d\}$. We also show a connection between the asymptotic growth of $β_n^d$ and the value of the Riemann zeta function $ζ(d)$. Finally, we show that $α_n^d$ arises naturally in the study of Mathieu-Zhao subspaces in products of finite fields.
