Galois groups of random additive polynomials

TL;DR

This paper investigates the distribution of Galois groups of random additive polynomials over finite fields, showing they are typically large and identifying conditions for specific subgroups to appear as the degree grows.

Contribution

It provides the first probabilistic analysis of Galois groups of random additive polynomials, establishing when they contain large subgroups like SL_n(q).

Findings

Galois groups are almost surely large as degree increases.

Necessary and sufficient conditions for Galois groups to contain SL_n(q).

Analysis of limits with respect to parameters q, n, and d.

Abstract

We study the distribution of the Galois group of a random $q$-additive polynomial over a rational function field: For $q$ a power of a prime $p$, let $f=X^{q^n}+a_{n-1}X^{q^{n-1}}+\ldots+a_1X^q+a_0X$ be a random polynomial chosen uniformly from the set of $q$-additive polynomials of degree $n$ and height $d$, that is, the coefficients are independent uniform polynomials of degree ${\rm deg}\, a_i\leq d$. The Galois group $G_f$ is a random subgroup of ${\rm GL}_n(q)$. Our main result shows that $G_f$ is almost surely large as $d,q$ are fixed and $n\to \infty$. For example, we give necessary and sufficient conditions so that ${\rm SL}_n(q)\leq G_f$ asymptotically almost surely. Our proof uses the classification of maximal subgroups of ${\rm GL}_n(q)$. We also consider the limits: $q,n$ fixed, $d\to \infty$ and $d,n$ fixed, $q\to \infty$, which are more elementary.