学术报告报告主题: Divisibility among power GCD matrices and power LCM matrices
报 告 人: 洪绍方教授（四川大学）
报告时间： 2025年9月29日（星期一）上午10:00-11:00
报告地点： 37号楼3A01报告厅
邀 请 人: 胡甦 副教授
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数学学院
2025年9月28日
报告摘要：
For any integers $x$ and $y$, let $(x, y)$ and $[x, y]$ stand for the greatest common divisor and the least common multiple of $x$ and $y$, respectively. 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 $(S^a)$ and $[S^a]$ the $n\times n$ matrix having the $a$th power of $(x_i,x_j)$ and $[x_i,x_j]$, respectively, as its $(i,j)$-entry. In 1992, Bourque and Ligh showed that if $S$ is factor closed (that is, $S$ contains all positive divisors of any element of $S$), then the GCD matrix $(S)$ divides the LCM matrix$[S]$ (written as $(S)|[S]$) in the ring $M_n({\mathbb Z})$ of $n\times n$ matrices over the integers. In 2008, Hong proved that $(S^a)|(S^b)$, $(S^a)|[S^b]$ and $[S^a]|[S^b]$ in the ring $M_{n}({\mathbb Z})$ when $a|b$ and $S$ is a divisor chain (namely, there is a permutation $\sigma$ of order $n$ such that $x_{\sigma(1)}|...|x_{\sigma(n)}$). In this talk, we show that if $a|b$ and $S$ is factor closed, then $(S^a)|(S^b)$, $(S^a)|[S^b]$ and $[S^a]|[S^b]$ in the ring $M_{n}({\mathbb Z})$. The proof is algebraic and $p$-adic. Our result extends the Bourque-Ligh theorem. Finally, some interesting conjectures are proposed.
报告人介绍：
洪绍方是四川大学数学学院教授、博士生导师。他是教育部新世纪优秀人才，也是四川省学术和技术带头人。洪绍方主要研究数论、算术几何和编码理论。他先后主持国家自然科学基金和教育部博士点基金等10多个纵向项目。在学术成果方面，已在国内外30多种重要数学期刊上发表论文100多篇，其中SCI收录论文80多篇。他还担任国际数学SCI期刊《AIMS Mathematics》和《Journal of Mathematics》等的编委。