论文标题
带有规定的零和子序列的序列的结构：排名两个$ p $ -groups
Structure of a sequence with prescribed zero-sum subsequences: Rank Two $p$-groups
论文作者
论文摘要
令$ g =（\ mathbb z/n \ mathbb z）\ oplus（\ mathbb z/n \ mathbb z）$。令$ \ mathsf {s} _ {\ leq k}（g）$为最小的整数$ \ ell $，使得$ g $的$ \ ell $项的每个序列都具有重复，允许使用，具有最多$ k $的非零零sum sequinence。众所周知，$ \ Mathsf {s} _ {\ leq 2n-1-k}（g）= 2n-1+k $ for $ k \ for $ k \ in [0，n-1] $，其极端序列的结构显示了此绑定的紧密结构，当$ k \ in \ in \ in \ in \ in \ in \ {0,1，n-1，n-1 \} $，以及各种特殊情况时，以及各种特殊情况时，以及当$ k \ n时$ k \] [2] $ [2，2。2，2。对于剩余的值$ k \在[2，n-2] $中，长度为$ 2n-2+k $的极端序列的表征避免了最多$ 2N-1-k $的非空的零长度，它一般仍保持打开状态，并指出他们必须所有人都必须具有$ e_1^{[n-1]} \ boldsymbol $ e_1^{[n-1]} \ boldsymbol {e_____] \ boldsymbol {\ cdot}（e_1 +e_2）^{[k]} $在某些基础上$（e_1，e_2）$ for $ g $。这里$ x^{[n]} $表示由$ x $重复$ n $ times的术语组成的序列。在本文中，当$ n $是PRIME时，我们为所有$ k \ in [2，n-2] $中的所有$ k \建立了这个猜想，鉴于其他最近的工作，这意味着所有等级两个Abelian群体的猜想结构。
Let $G=(\mathbb Z/n\mathbb Z) \oplus (\mathbb Z/n\mathbb Z)$. Let $\mathsf {s}_{\leq k}(G)$ be the smallest integer $\ell$ such that every sequence of $\ell$ terms from $G$, with repetition allowed, has a nonempty zero-sum subsequence with length at most $k$. It is known that $\mathsf {s}_{\leq 2n-1-k}(G)=2n-1+k$ for $k\in [0,n-1]$, with the structure of extremal sequences showing this bound tight determined when $k\in \{0,1,n-1\}$, and for various special cases when $k\in [2,n-2]$. For the remaining values $k\in [2,n-2]$, the characterization of extremal sequences of length $2n-2+k$ avoiding a nonempty zero-sum of length at most $2n-1-k$ remained open in general, with it conjectured that they must all have the form $e_1^{[n-1]} \boldsymbol{\cdot} e_2^{[n-1]} \boldsymbol{\cdot} (e_1 +e_2)^{[k]}$ for some basis $(e_1,e_2)$ for $G$. Here $x^{[n]}$ denotes a sequence consisting of the term $x$ repeated $n$ times. In this paper, we establish this conjecture for all $k\in [2,n-2]$ when $n$ is prime, which in view of other recent work, implies the conjectured structure for all rank two abelian groups.
