The Galois group of random reciprocal polynomials

Let $P = a_m T^m + \sum_{j=0}^{m-1} a_j (T^{m-j} + T^{m+j})$ be a monic reciprocal polynomial of even degree n, with the a_j drawn independently and uniformly at random from {1, 2, …, H} for some fixed integer H ≥ 35. In joint work with Dimitris Koukoulopoulos, we show that P is irreducible over the rationals, and has large Galois group, with probability tending to 1 as n tends to infinity. This agrees, in a setting of coefficients with ‘mild’ dependencies, with the long-helf belief going back in various forms to Hilbert, Van der Waerden, Odlyzko―Poonen and Konyagin that a typical polynomial is irreducible and has Galois group as large as possible. I will discuss some aspects of the proof, which draws from classical polynomial theory, Fourier analysis, probability theory and group theory, and explain why the dependence of the coefficients of P does (not) matter.