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arXiv 2407.07644math.CO
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$\u2124_p^d$中零和环的改进界

Improved bounds for zero-sum cycles in $\mathbb{Z}_p^d$

Micha Christoph, Charlotte Knierim, Anders Martinsson, Raphael Steiner

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AI总结：

本文研究素数阶循环群幂空间$\u2124_p^d$的零和环参数上界问题，证明p=2时n(Z₂ᵈ)≤5d、p≥3时n(Zₚᵈ)≤O(pd log d)，还推广了拟阵框架下的超图匹配结果。

摘要：

对于有限阿贝尔群$(Γ,+)$，令$n(Γ)$表示满足以下条件的最小正整数$n$：对阶为$n$的完全有向图的弧用$Γ$中元素进行任意标记，都存在一个有向环，使得环上所有弧标记的总和等于$0$。Alon和Krivelevich率先研究了循环群上的参数$n(\cdot)$，并证明$n(\mathbb{Z}_q)=O(q\log q)$。在研究$Γ=\mathbb{Z}_p^d$（素数阶循环群的幂）这一典型情形时，Letzter和Morrison近期证明了$n(\mathbb{Z}_p^d) \le O(pd(\log d)^2)$，且$n(\mathbb{Z}_2^d)\le O(d \log d)$，随后他们提出了证明对所有素数$p$和自然数$d$，$n(\mathbb{Z}_p^d)\le O(pd)$这一（渐近最优）上界的问题。本文解决了$p=2$时的该问题，并改进了所有素数$p\ge3$时的界：证明了$n(\mathbb{Z}_2^d)\le 5d$，且$n(\mathbb{Z}_p^d)\le O(pd\log d)$。第一个界确定了$n(\mathbb{Z}_2^d)$的误差不超过5倍乘数因子，第二个界的紧性差距仅为$\log d$因子。此外，我们的结果表明，若关于$\mathbb{Z}_p^d$中加法基的Jaeger-Linial-Payan-Tarsi著名猜想的强形式成立，则对任意$p$和$d$，都有紧界$n(\mathbb{Z}_p^d)=Θ(pd)$。\n在证明这些结果的过程中，我们在拟阵框架下推广了Haxell的超图匹配结果。具体而言，我们得到了超图中存在匹配的充分条件：该超图的超边由拟阵元素标记，且匹配中的边能诱导出拟阵的一个基。我们认为这些结论本身具有独立的研究价值。

英文摘要：

For a finite Abelian group $(Γ,+)$, let $n(Γ)$ denote the smallest positive integer $n$ such that for each labelling of the arcs of the complete digraph of order $n$ using elements from $Γ$, there exists a directed cycle such that the total sum of the arc-labels along the cycle equals $0$. Alon and Krivelevich initiated the study of the parameter $n(\cdot)$ on cyclic groups and proved that $n(\mathbb{Z}_q)=O(q\log q)$. Studying the prototypical case when $Γ=\mathbb{Z}_p^d$ is a power of a cyclic group of prime order, Letzter and Morrison recently showed that $n(\mathbb{Z}_p^d) \le O(pd(\log d)^2)$ and that $n(\mathbb{Z}_2^d)\le O(d \log d)$. They then posed the problem of proving an (asymptotically optimal) upper bound of $n(\mathbb{Z}_p^d)\le O(pd)$ for all primes $p$ and $d \in \mathbb{N}$. In this paper, we solve this problem for $p=2$ and improve their bound for all primes $p \ge 3$ by proving $n(\mathbb{Z}_2^d)\le 5d$ and $n(\mathbb{Z}_p^d)\le O(pd\log d)$. While the first bound determines $n(\mathbb{Z}_2^d)$ up to a multiplicative error of $5$, the second bound is tight up to a $\log d$ factor. Moreover, our result shows that a tight bound of $n(\mathbb{Z}_p^d)=Θ(pd)$ for arbitrary $p$ and $d$ would follow from a (strong form) of the well-known conjecture of Jaeger, Linial, Payan and Tarsi on additive bases in $\mathbb{Z}_p^d$. Along the way to proving these results, we establish a generalization of a hypergraph matching result by Haxell in a matroidal setting. Concretely, we obtain sufficient conditions for the existence of matchings in a hypergraph whose hyperedges are labelled by the elements of a matroid, with the property that the edges in the matching induce a basis of the matroid. We believe that these statements are of independent interest.

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