Irreducibility and Galois groups of random reciprocal polynomials of large degree

TL;DR

This paper proves that large degree random reciprocal polynomials are irreducible with high probability and their Galois groups are typically the full hyperoctahedral group or its index-2 subgroups, under broad conditions.

Contribution

It establishes irreducibility and Galois group distribution for random reciprocal polynomials with coefficients from broad probability measures, extending previous work on non-reciprocal polynomials.

Findings

Polynomials are irreducible with probability ≥ 1 - Cm^{-c}.

Galois groups are typically the full hyperoctahedral group or its index-2 subgroups.

Results hold for broad classes of coefficient distributions satisfying Fourier conditions.

Abstract

Let $A = a_0T^m + \sum_{j=1}^{m-1} a_j (T^{m-j}+T^{m+j}) + T^{2m}+1 \in \mathbf{Z}[T]$ 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 $\ge 1-Cm^{-c}$ 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 $a_0 \neq 0$. 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.