论文标题
关于Brian Alspach的猜想的一些新结果
Some new results about a conjecture by Brian Alspach
论文作者
论文摘要
在本文中，我们考虑由Brian Alspach提出的以下猜想，涉及有限环状组中的部分总和：给定一个子集$ a $ a $ a $ \ mathbb {z} _n \ setMinus \ setMinus \ {0 \ {0 \} $ size $ k $的$ $（a_1，\ ldots，a_k）$ a $的元素$，使得部分总和$ s_i = \ sum_ = \ sum_ {j = 1}^i a_j $，$ i = 1，\ ldots，k $，nonzero and nonzero and pairywise独特。已知该猜想对于$ k \ leq 11 $的循环序列组的子集已知。在这里，我们将这些结果扩展到任何无扭转的Abelian组，因此，我们在$ \ Mathbb {z} _n $中提供了渐近结果。 我们还考虑了一个相关的猜想，最初由Ronald Graham提出：给定一个子集$ a $ a $ a $ \ mathbb {z} _p \ setminus \ {0 \} $，其中$ p $是质量，其中存在$ a $的元素的订购，使部分总和是不同的。使用Hicks，Ollis和Schmitt开发的方法，基于Alon的组合无效的Nullstellensatz，我们证明了这种猜想对子集的有效性$ a $ a $ a $ a $ a a $ a $ of $ 12 $。
In this paper we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset $A$ of $\mathbb{Z}_n\setminus \{0\}$ of size $k$ such that $\sum_{z\in A} z\not= 0$, it is possible to find an ordering $(a_1,\ldots,a_k)$ of the elements of $A$ such that the partial sums $s_i=\sum_{j=1}^i a_j$, $i=1,\ldots,k$, are nonzero and pairwise distinct. This conjecture is known to be true for subsets of size $k\leq 11$ in cyclic groups of prime order. Here, we extend such result to any torsion-free abelian group and, as a consequence, we provide an asymptotic result in $\mathbb{Z}_n$. We also consider a related conjecture, originally proposed by Ronald Graham: given a subset $A$ of $\mathbb{Z}_p\setminus\{0\}$, where $p$ is a prime, there exists an ordering of the elements of $A$ such that the partial sums are all distinct. Working with the methods developed by Hicks, Ollis and Schmitt, based on the Alon's combinatorial Nullstellensatz, we prove the validity of such conjecture for subsets $A$ of size $12$.
