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

TL;DR

This paper investigates the Galois groups of reciprocal polynomials, revealing that a significant proportion have smaller than maximal Galois groups, and extends previous results on polynomial Galois groups with new insights into their structure.

Contribution

It provides a quantitative analysis of Galois groups of reciprocal polynomials, answering a longstanding question and extending Bhargava's recent work on polynomial Galois groups.

Findings

Proportion of reciprocal polynomials with smaller Galois groups is bounded by constants times H^{-1} log H.

Main contribution from polynomials where (-1)^n f(1) f(-1) is a square.

Galois groups often lie in an index-2 subgroup, not the full symmetric group.

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 $S_2 \wr S_n$ 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-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.