Divisibility among power
GCD matrices and power LCM matrices
on certain gcd-closed sets
报告专家：朱光艳 副教授 （湖北民族大学）
报告时间：12月26日（周五）9：30-10：30
报告地点：四川大学数学学院西303报告厅
报告摘要：
Let $(x, y)$ and $[x, y]$ denote the greatest common divisor and the least common multiple of the integers $x$ and $y$ respectively. We denote by $|T|$ the number of elements ofa finite set $T$. Let $a,b$ and $n$ be positive integers and let $S=\{x_1, ..., x_n\}$ be a set of $n$ distinct positive integers. We denote by $(S^a)$ (resp. $[S^a]$) the $n\times n$ matrix whose $(i,j)$-entry is $(x_i,x_j)^a$ (resp. $[x_i,x_j]^a$). For any $x\in S$, let $G_{S}(x)$ stand for the set of all the greatest-type divisors of $x$ in $S$, that is, $G_{S}(x):=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S)\Rightarrow y\in \{d,x\}\}$. We say that the set $S$ satisfies the condition $\mathcal{G}$ if for any $x\in S$, either $|G_S(x)|\le 1$, or $|G_S(x)|\ge 2$ and $[y_1,y_2]=x$ as well as $(y_1,y_2)\in G_S(y_1)\cap G_S(y_2)$ for any $\{y_1,y_2\}\subseteq G_S(x)$. In 1995, Bourque and Ligh showed that $(S^a)$ divides $[S^a]$ (written as $(S^a)\mid [S^a]$) in the ring $M_n(\bf Z)$ of $n\times n$ matrices over the integers if $S$ is factor closed (i.e. $d\in S$ if $d|x$ for any $x\in S$). In 2025, Hong proved that any factor-closed set is a gcd-closed set (namely, $(x_i, x_j)\in S$ for all integers $i$ and $j$ with $1\le i, j\le n$) satisfying the condition $\mathcal{G}$, and conjectured that for arbitrary positive integers $a$ and $b$ with $a|b$, if $S$ is a gcd-closed set satisfying the condition $\mathcal{G}$, then $(S^a)\mid(S^b)$, $(S^a)\mid[S^b]$ and $[S^a]\mid[S^b]$ in the ring $M_n(\bf Z)$.
In this talk, we partially confirm Hong's conjecture for the case when $\max_{x\in S}\{|G_S (x)|\}\le 3$. This result extends the Feng-Hong-Zhao theorem obtained in 2009 and the Chen-Hong-Zhao theorem gotten in 2022. Furthermore, we show the existence of gcd-closed sets $S$ such that $S$ does not satisfy the condition $\mathcal{G}$ and such factorizations are true.
This is a joint work with J.X. Wan
专家简介：
朱光艳，湖北民族大学副教授，硕士生导师。于2020年9月-2024年6月，在四川大学数学学院攻读博士学位，获得理学博士学位。目前已在Forum Math., Bull. Aust. Math. Soc., Colloq. Math., Proc. Amer. Math. Soc.,Inter. J. Number Theory和Discrete Math.等国际数学期刊上发表论文十多篇。主要研究方向为数论及其应用。主持或完成多项省级科研或教改项目。
邀请人：洪绍方
