Irreducibility and Galois groups of random reciprocal polynomials of large degree

Abstract

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 constants $c,C>0$. In addition, we prove that with the same probability the Galois group ${\mathcal{G}}_A$ of $A$ is either the full hyperoctahedral group $C_2\wr S_m$ 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 $a_0+\cdots +a_{m-1}{T}^{m-1}+T^m$, conditioning on $a_0\ne 0$, is irreducible and has as Galois group either the symmetric group $S_m$ or the alternating group $A_m$ with high probability. For reciprocal polynomials, we can study the discriminant of $A$ and rule out with high probability that ${\mathcal{G}}_A\le (C_2\wr S_m)\cap A_{2m}$, i.e., that ${\mathcal{G}}_A$ is contained in the maximal alternating subgroup. Furthermore, we establish a bound on the probability that ${\mathcal{G}}_A\le C_2\times S_m$ by examining the Frobenius at suitable primes. In the process, we establish a Łuczak--Pyber theorem for the group $C_2\wr S_m$, which may be of independent interest.

v2: 49 pages; minor corrections