Irreducibility and Galois groups of random reciprocal polynomials of large degree

Let $A=a_0{T}^m+{\sum }_{j=1}^{m-1}{a}_j(T^{m-j}+T^{m+j})+T^{2m}+1\in \mathbf{Z}[T]$ be a monic reciprocal polynomial of degree $2m$ sampled randomly by selecting its coefficients $a_0,a_1,\dots ,a_{m-1}$ independently according to a given probability measure $\mu$ on $\mathbf{Z}$. For a wide range of measures $\mu$, we prove that $A$ is irreducible with probability $\ge 1-C{m}^{-c}$ for some absolute constants $c,C>0$. In addition, we prove that with the same probability the Galois group of $A$ is either the full hyperoctahedral group ${\mathcal{C}}_2\wr {\mathcal{S}}_m$ or one of two of its index-$2$ subgroups. The main condition that $\mu$ must satisfy is of Fourier-theoretic nature, and holds for example when $\mu$ is the uniform measure on a set of at least $35$ consecutive integers, or on an arbitrary, sufficiently large subset of an interval $[-H,H]$, with $H$ larger than some absolute constant. Our most general result allows for each $a_j$ to be sampled by its own probability measure ${\mu }_j$. Our approach builds on earlier work of Bary-Soroker, Kozma and the second author, who proved for essentially the same ${\mu }_j$ that the ‘standard’ monic polynomial $a_0+\cdots +a_{m-1}{T}^{m-1}+T^m$ is irreducible and has as Galois group either the symmetric group ${\mathcal{S}}_m$ or the alternating group ${\mathcal{A}}_m$ with high probability, conditioning on $a_0\mathrm{\ne }0$. In our setting of reciprocal polynomials, we can rule out (all subgroups of) the maximal alternating subgroup $({\mathcal{C}}_2\wr {\mathcal{S}}_m)\cap {\mathcal{A}}_{2m}$ of the hyperoctahedral group as likely Galois group of $A$ by analyzing its discriminant.