Galois groups of reciprocal polynomials and the van der Waerden–Bhargava theorem
Abstract
We study the Galois groups G f of degree 2n 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.
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Polynomial and algebraic computation
Advanced Mathematical Identities
Publication details
- Journal / source
- Algebra & Number Theory
- Volume
- 20
- Issue
- 3
- Pages
- 603–628
- Publication date
- 24 March 2026
- Language
- EN