Galois groups of random additive polynomials

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.