﻿ 【学术讲座】On the irreducibility and Galois group of summation of exponential Taylor polynomials-西南交通大学数学学院

# 【学术讲座】On the irreducibility and Galois group of summation of exponential Taylor polynomials

2023-09-13  点击：[]

报告时间：2023年9月14日下午15:30-16:30

报告地点：西南交通大学犀浦校区7教7510

报告人：洪绍方

摘要： Let $n\ge 1$ be an integer and $f(x)=\frac{x^n}{n!}+\sum_{i=0}^{n-1}c_i\frac{x^i}{i!}$,where $c_0,c_1,...,c_{n-1}$ are arbitrary integers. In this talk, we show that if $f(x)$ is reducible over $\Q$, then there exists an irreducible factor whose degree is less than the maximal prime divisor of $c_0$. We also obtain all the possible degree of $f(x)$ which is reducible over $\Q$ when all the prime factor of $c_0$ is a subset of $\{2,3,5\}$. This extends a theorem of I. Schur. Let $p\in \{2,3,5\}$ and let $e_{n}(x):=\sum_{i=0}^n\frac{x^i}{i!}$ denote the truncated exponential Taylor polynomial and $\E_{n,p}(x):=e_n(x)+(p-1)e_{n-1}(x)$. We prove that $\E_{n,p}(x)$ is irreducible if $(n,p)\not\in\{(2,2),(4,2)\}$. Furthermore, we show that the Galois group ${\rm Gal}_{\Q}(\E_{n,p})$ contains $A_{n}$ except for $(n,p)=(4,2)$, in which case, ${\rm Gal}_{\Q}(\E_{4,2})=S_3$. Finally, we show that the Galois group ${\rm Gal}_{\Q}(\E_{n,2})$ is $S_n$ if $n\equiv 3 \pmod 4$, or if $n$ is even and $v_q(n!)$ is odd for a prime divisor $q$ of $n-1$, or if $n\equiv 1\pmod 4$ and $n-2$ equals the product of an odd prime number $l$ which is coprime to $\sum_{i=1}^{l-1}2^{l-1-i}i!$ and a positive integer coprime to $l$. This is a joint work with Dr. L.F. Ao.

报告人简介：洪绍方，现任四川大学数学学院教授，博士生导师。教育部新世纪优秀人才，四川省学术和技术带头人。主要研究方向：数论，算术几何和编码理论。已经在国内外数学期刊上发表学术论文百余篇，已经培养毕业硕士、博士几十名。

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