On the Galois Theory of Generalized Laguerre Polynomials and Trimmed Exponential

TL;DR

This paper investigates the Galois groups of certain generalized Laguerre polynomials and trimmed exponentials, showing that they are asymptotically almost surely the full symmetric group for a wide range of parameters.

Contribution

It establishes that the Galois groups of these polynomials are generically the full symmetric group, extending understanding of their algebraic properties.

Findings

Galois groups are almost surely the full symmetric group for the studied polynomials.

Results hold for parameters up to subpolynomial size in x.

Provides new insights into the algebraic structure of generalized Laguerre polynomials.

Abstract

Inspired by the work of Schur on the Taylor series of the exponential and Laguerre polynomials, we study the Galois theory of trimmed exponentials $f_{n,n+k}=\sum_{i=0}^{k} \frac{x^{i}}{(n+i)!}$ and of the generalized Laguerre polynomials $L^{(n)}_k$ of degree $k$. We show that if $n$ is chosen uniformly from $\{1,\ldots, x\}$, then, asymptotically almost surely, for all $k\leq x^{o(1)}$ the Galois groups of $f_{n,n+k}$ and of $L_{k}^{(n)}$ are the full symmetric group $S_k$.