Irreducibility and Galois groups of random reciprocal polynomials of large degree

TL;DR

This work analyzes irreducibility and Galois groups of random monic reciprocal polynomials of degree $2m$ with coefficients sampled from a broad class of probability measures satisfying a Fourier-type condition. Building on Bary-Soroker–Kozma–Koukoulopoulos, the authors develop a reciprocal-polynomial framework via the trace polynomial $A_{\mathsf{R}}$, establish strong (modulo primes) equidistribution results, and combine probabilistic bounds (both $L^\infty$ and $L^1$) with discriminant analysis to exclude obstructive Galois-subgroups. They prove that $A$ is irreducible with probability $1-O(m^{-c})$ and that its Galois group is, with high probability, either the full hyperoctahedral group $\mathcal{C}_2\wr\mathcal{S}_m$ or one of two index-$2$ subgroups, with the remaining small case ruled out by discriminant considerations. The argument hinges on a detailed understanding of the divisor structure of reciprocal polynomials, a careful treatment of dependencies among coefficients, and a Fourier-analytic approach to joint factorisation modulo several primes, all culminating in a robust master theorem. The results extend previous nonreciprocal-model work to the reciprocal setting and include highly general coefficient measures, yielding unconditional irreducibility rates and Galois-group classifications in a sparse but structurally rigid polynomial family.

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 $μ$ on $\mathbf{Z}$. For a wide range of measures $μ$, 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 $μ$ must satisfy is of Fourier-theoretic nature, and holds for example when $μ$ 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 $μ_j$. Our approach builds on earlier work of Bary-Soroker, Kozma and the second author, who proved for essentially the same $μ_j$ 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.

Theorems & Definitions (130)

• Definition 1.1: Reciprocal polynomial
• Remark 1.2
• Theorem A
• Definition 1.3
• Theorem B
• Theorem C
• Remark 1.4
• Lemma 2.1: Divisors of reducible reciprocal polynomials
• Proposition 2.2: Prime factorization of reciprocal polynomials
• Proposition 2.3: Small degree factors