Galois groups of random additive polynomials (Q6567136)

This work follows recent breakthroughs on Galois groups of random polynomials, especially over the integers. In number theory, there is an analogy between number fields and function fields, e.g., \(\mathbb Q \leftrightarrow \mathbb F_q(t)\). With this in mind, the authors consider polynomials with coefficients in \(\mathbb F_q[t]\). They restrict attention to additive polynomials, namely those of the shape\N\[\Nf(X) = X^{q^n} + a_{n-1}(t) X^{q^{n-1}} + \cdots + a_1(t) X^q + a_0(t) X.\N\]\NIf \(a_0(t), \ldots, a_{n-1}(t)\) are uniformly random polynomials of degree at most \(d\) over \(\mathbb F_q\), then \(f\) is a random additive polynomial over \(\mathbb F_q[t]\). The Galois group \(G_f\) of \(f\) over \(\mathbb F_q(t)\) acts on the roots of \(f\) so, up to conjugacy, we can regard it as a subgroup of \(\mathrm{GL}_n(\mathbb F_q)\). The main objective is to show that \(G_f = \mathrm{GL}_n(\mathbb F_q)\) asymptotically almost surely as \(n \to \infty\), with \(d,q\) fixed. One could send \(d\) or \(q\) to infinity instead, but the authors explain that the task is then easier. They achieve their goal after excluding a natural obstruction.