主 讲 人 ：Bhawesh Mishra    博士后

活动时间：2025-06-18 19:00:00

地      点 ：理科群1号楼D204室 https://us06web.zoom.us/j/86763384947?pwd=qXhOzOcHvaaiw2jqADI8iqNavdgm14.1 会议 ID: 86763384947；密码: 612593

主办单位：数学科学学院

讲座内容：

Let $q$ be an odd prime. Consider a finite set $S$ of integers that contains a $q$th power modulo (almost) every prime but does not contain an integer $q$th power. In other words, $S$ is such that the polynomial $\prod_{s \in S} (x^q - s)$ has local roots (almost) everywhere but fails to have a global root. Such sets $S$ constitute counterexamples to the local-global principle in number theory. More generally, let $k$ be a natural number and $\Omega$ be the arithmetic function that counts prime factors with multiplicity. We will consider finite sets $S$ of integers that contain a $q$th power modulo (almost) every integer $N$ with $\Omega(N) \leq k$ yet fail to contain an integer $q$th power. Let us denote the collection of such sets $S$ by $\mathcal{T}_{k,q}$.

On the other hand, finite geometers and coding theorists are interested in what is known as blocking sets. Given integers $n \geq k \geq 1$, a subset $\mathcal{S}$ of the projective space $\mathrm{PG}(\mathbb{F}_q^n)$ is said to be a k-blocking set if, for every subspace $W$ of $\mathbb{F}_q^n$ with codimension $k$, we have $\mathrm{PG}(W) \cap \mathcal{S} \neq \emptyset$.

In this talk, we will bridge the fields of number theory and finite geometry by showing that sets in the collection $\mathcal{T}_{k,q}$ are in one-to-one correspondence with $k$-blocking sets. More specifically, a finite subset $S$ of integers belongs to $\mathcal{T}_{k, q}$ if and only if the set of projective points associated with $S$ forms a $k$-blocking set in $\mathrm{PG}(\mathbb{F}_q^n)$, where $n$ is the number of primes $p$ such that $p$ divides the $q$-free part of an element of $S$. Using this correspondence, we can obtain several number-theoretic consequences, such as a lower bound on the cardinality of such sets $S$, and precise information about the multiplicative structure of the elements of $S$ that would otherwise be unavailable. Time permitting, we will also discuss some directions for future research on this topic.

The talk will largely be based on a joint work with Paolo \textsc{Santonastaso} from Universit\`a degli Studi della Campania "Luigi Vanvitelli" (Caserta, Italy).

主讲人介绍：

The speaker hails from Nepal. He obtained a PhD in mathematics from the Ohio State University (Columbus, USA) in 2023. Since August 2023, he has been working in the capacity of a postdoctoral fellow in the Department of Mathematical Sciences at the University of Memphis (Memphis, USA). His research interests lie at the interface of number theory and algebra but also have parallel connections to finite geometry and combinatorics. He also enjoys reading, playing soccer, and attending classical music concerts.