Random polynomials with integer coefficients tend to be irreducible and to have a large Galois group with high probability. This was shown more than a century ago in the large box model, where we choose the coefficients uniformly from a box and let its size go to infinity, while only recently there are results in the restricted box model, when the size of the box is bounded and its dimension (i.e., the degree of the polynomial) goes to infinity. In this talk, we will discuss an important class of random polynomials: additive polynomials, which have coefficients in the polynomial ring over a finite field. In this case, the roots form a vector space, hence the Galois group is naturally a subgroup of GLn. While we prove that the Galois group is the full matrix group both in the large box model, and in the large finite field limit, our main result is in the restricted box model: under some necessary condition the Galois group is large (in the sense that it contains SLn) asymptotically almost surely, as the degree goes to infinity. The proof relies crucially on deep results on subgroups of GLn by Fulman and Guralnick, combined with tools from algebra and number theory.
Based on joint work with Alexei Entin and Eilidh McKemmie.
This video is part of the Number Theory Web Seminar series.
