Galois groups of reciprocal polynomials and the van der Waerden–Bhargava theorem

Anderson, Theresa C.; Bertelli, Adam; O’Dorney, Evan M.

doi:10.2140/ant.2026.20.603

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Galois groups of reciprocal polynomials and the van der Waerden–Bhargava theorem

Theresa C. Anderson, Adam Bertelli and Evan M. O’Dorney

Vol. 20 (2026), No. 3, 603–628

DOI: 10.2140/ant.2026.20.603

Abstract

We study the Galois groups $G_{f}$ of degree $2 n$ reciprocal (a.k.a. palindromic) polynomials $f$ of height at most $H$ , finding that $G_{f}$ falls short of the maximal possible group $S_{2} ≀ S_{n}$ for a proportion of all $f$ bounded above and below by constant multiples of $H^{- 1} \log H$ , whether or not $f$ is required to be monic. This answers a 1998 question of Davis, Duke and Sun and extends Bhargava’s 2023 resolution of van der Waerden’s 1936 conjecture on the corresponding question for general polynomials. Unlike in that setting, the dominant contribution comes not from reducible polynomials but from those $f$ for which ${(- 1)}^{n} f (1) f (- 1)$ is a square, causing $G_{f}$ to lie in an index- $2$ subgroup.

Keywords

arithmetic statistics, van der Waerden's conjecture, reciprocal polynomial, geometric sieve

Mathematical Subject Classification

Primary: 11R32, 11R45, 11C08, 11N35, 20E22

Milestones

Received: 17 July 2024

Revised: 16 January 2025

Accepted: 3 March 2025

Published: 24 March 2026

Authors

Theresa C. Anderson
Carnegie Mellon University Pittsburgh, PA United States

Adam Bertelli
Pennsylvania State University University Park, PA United States

Evan M. O’Dorney
Carnegie Mellon University Pittsburgh, PA United States

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Vol. 20, No. 3, 2026