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Number Theory Lunch Seminar
Counting integral matrices with a given characteristic polynomial
Counting integral matrices with a given characteristic polynomial
Organizers
Speaker
Time
Thursday, December 11, 2025 12:15 PM - 1:00 PM
Venue
A4-1
Abstract
Let $P(x)\in \mathbb{Z}[x]$ be a monic irreducible polynomial of degree $n$ and $M_n(\mathbb{Z})$ be the space of $n\times n$ integral matrices.
Let $V=\{X\in M_n(\mathbb{Z}): \det(xI-X)=P(x)\}$ and $B_T$ be the Euclidean ball centered at $0$ of radius $T$ in $M_n(\mathbb{R})$.
In this talk, I will explain the asymptotic formula of Eskin, Mozes and Shah:
\[
\lim_{T\rightarrow \infty}\frac{\# (V\cap B_T)}{T^{n(n-1)/2}}=C_P
\]
for some constant $C_P>0$. If time permits, I will explain the interpretation of the constant $C_P$ in terms of orbital integrals by work of Yuchan Lee base on work of Dasheng Wei and Fei Xu. No new results will be discussed in this talk.
Let $V=\{X\in M_n(\mathbb{Z}): \det(xI-X)=P(x)\}$ and $B_T$ be the Euclidean ball centered at $0$ of radius $T$ in $M_n(\mathbb{R})$.
In this talk, I will explain the asymptotic formula of Eskin, Mozes and Shah:
\[
\lim_{T\rightarrow \infty}\frac{\# (V\cap B_T)}{T^{n(n-1)/2}}=C_P
\]
for some constant $C_P>0$. If time permits, I will explain the interpretation of the constant $C_P$ in terms of orbital integrals by work of Yuchan Lee base on work of Dasheng Wei and Fei Xu. No new results will be discussed in this talk.
Speaker Intro
Dr. DENG Taiwang has joined BIMSA in November 2022 as an Assistant Professor. His research interests are in the Langlands program (broadly speaking, the arithmetic, analytic and representation aspects of it). He obtained a Phd in Mathematics from the University of Paris 13. Previously, he has held the postdoctorial positions in Bonn University, the Max Planck Institute of Mathematics in Bonn and Tsinghua University.