Irreducibility and Galois groups of random reciprocal polynomials of large degree

2025/10/21 by David Hokken, Dimitris Koukoulopoulos, Hokken, David +1
Mathematics · #Geometry and complex manifolds #Algebraic Geometry and Number Theory #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2510.18857

Abstract

Let A = a₀T^m + ∑_j=1^m-1 a_j (T^m-j+T^m+j) + T^2m+1 ∈ Z[T] be a monic reciprocal polynomial of degree 2m sampled randomly by selecting its coefficients a₀,a₁,…,a_m-1 independently according to a given probability measure μ on Z. For a wide range of measures μ, we prove that A is irreducible with probability ≥ 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 C₂ \wr 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 a₀ + ⋯ + a_m-1T^m-1 + T^m is irreducible and has as Galois group either the symmetric group S_m or the alternating group A_m with high probability, conditioning on a₀ ≠ 0. In our setting of reciprocal polynomials, we can rule out (all subgroups of) the maximal alternating subgroup (C₂ \wr S_m) ∩ A_2m of the hyperoctahedral group as likely Galois group of A by analyzing its discriminant.

Citations

• Characterizing the support of semiclassical measures for higher-dimensional cat maps
• Galois groups of reciprocal polynomials and the van der Waerden-Bhargava theorem
• Full Galois groups of polynomials with slowly growing coefficients
• Galois groups of random polynomials over the rational function field
• Irreducibility of Littlewood polynomials of special degrees
• Counting (skew-)reciprocal Littlewood polynomials with square discriminant
• Probabilistic Galois Theory -- The Square Discriminant Case
• Galois groups of random integer polynomials and van der Waerden's Conjecture
• Towards van der Waerden's conjecture
• Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial
• Sums of two squares in short intervals in polynomial rings over finite fields

Related

• Galois Groups of Generic Polynomials
• Irreducibility of random polynomials of ℤ[X]
• Probabilistic Galois Theory in Function Fields
• Galois groups of random polynomials over the rational function field
• On the probability of irreducibility of random polynomials with integer coefficients