Algebraic properties of summation of exponential Taylor polynomials

TL;DR

This paper investigates the algebraic and Galois group properties of sums of truncated exponential polynomials, extending Schur's theorems and characterizing their irreducibility and Galois groups for various n.

Contribution

It extends Schur's results by analyzing the irreducibility and Galois groups of sums of exponential Taylor polynomials, providing new conditions and classifications.

Findings

$rac{x^n}{n!}+ ext{integer coefficients}$ polynomials are irreducible under certain conditions.

$ ext{E}_n(x)$ is irreducible for all $n$ except 2 and 4.

Galois groups of $ ext{E}_n(x)$ are mostly $A_n$ or $S_n$, with specific exceptions.

Abstract

Let $n\ge 1$ be an integer and $e_n(x)$ denote the truncated exponential Taylor polynomial, i.e. $e_{n}(x)=\sum_{i=0}^n\frac{x^i}{i!}$. A well-known theorem of Schur states that the Galois group of $e_n(x)$ over $\Q$ is the alternating group $A_n$ if $n$ is divisible by 4 or the symmetric group $S_n$ otherwise. In this paper, we study algebraic properties of the summation of two truncated exponential Taylor polynomials $\E_n(x):=e_n(x)+e_{n-1}(x)$. We show that $\frac{x^n}{n!}+\sum_{i=0}^{n-1}c_i\frac{x^i}{i!}$ with all $c_i \ (0\le i\le n-1)$ being integers is irreducible over $\Q$ if either $c_0=\pm 1$, or $n$ is not a positive power of $2$ but $|c_0|$ is a positive power of 2. This extends another theorem of Schur. We show also that $\E_n(x)$ is irreducible if $n\not\in\{2,4\}$. Furthermore, we show that ${\rm Gal}_{\Q}(\E_n)$ contains $A_{n}$ except for $n=4$, in which case, ${\rm Gal}_{\Q}(\E_4)=S_3$. Finally, we show that the Galois group ${\rm Gal}_{\Q}(\E_n)$ is $S_n$ if $n\equiv 3 \pmod 4$, or if $n$ is even and $v_p(n!)$ is odd for a prime divisor of $n-1$, or if $n\equiv 1\pmod 4$ and $n-2$ equals the product of an odd prime number $p$ which is coprime to $\sum_{i=1}^{p-1}2^{p-1-i}i!$ and a positive integer coprime to $p$.