Long sequences having no two nonempty zero-sum subsequences of distinct lengths
报告人:高维东教授 南开大学 时间:2020年7月10日19:30-20:30
摘要: Let $G$ be an additive finite abelian group. By $\mathrm{disc}(G)$ we denote the smallest positive integer $t$ such that every sequence $S$ over $G$ of length $|S|\geq t$ has two nonempty zero-sum subsequences of distinct lengths. We first extend the list of the groups $G$ for which $\mathrm{disc}(G)$ being known. Then we focus on the inverse problems on $\mathrm{disc}(G)$. Let $\mathcal {L}_1(G)$ denote the set of all positive integers $t$ with the property that, there is a sequence $S$ over $G$ with length $|S|=\mathrm{disc}(G)-1$ such that, every nonempty zero-sum subsequence of $S$ has the same length $t$. We determine $\mathcal {L}_1(G)$ for some special groups including the groups with large exponents compare to $|G|$, the groups of rank at most two, the groups $C_{p^{n}}^{r}$ with $3\leq r\leq p$ and the groups $C_{mp^{n}}\oplus H$, where $H$ is a $p$-group with $\mathsf D(H)\leq p^{n}$, and $\mathsf D(H)$ denotes the Davenport constant of $H$. In particular, we find some groups $G$ with $|\mathcal{L}_1(G)|\geq 2$ which disprove a recent conjecture in this area. Let $S$ be a sequence over $G$ such that all nonempty zero-sum subsequences have the same length. We determine the structure of $S$ for the cyclic group $C_n$ when $|S|\geq n+1$, and for the group $C_n\oplus C_n$ when $|S|=3n-2=\mathrm{disc}(C_n\oplus C_n)-1$.
简介:高维东教授1988年在东北师范大学获得硕士学位, 1994年从四川大学获得博士学位。 曾于大连理工大学及奥地利Graz大学从事博士后工作。现为南开大学组合数学中心教授。其主要从事组合数论研究,已发表研究论文100多篇。他建立了组合数论中两个重要不变量Erdos-Ginburg-Ziv定理常数与Davenport常数之间的一个基本的联系,从而将原本围绕它们各自独立进行的研究统一起来。