Galois groups of reciprocal polynomials II: Twisted reciprocal polynomials

TL;DR

This paper investigates the Galois groups of a family of twisted reciprocal polynomials, showing that the probability of not having the full hyperoctahedral group as Galois group diminishes at a specific rate as polynomial height increases.

Contribution

It extends previous work on reciprocal polynomials to twisted cases, establishing the asymptotic probability distribution of their Galois groups.

Findings

Probability that G_f is not the full hyperoctahedral group is Θ(H^{-1} log H)

The leading-order Galois group G_1 has index 2

Results are independent of the twisting parameter b

Abstract

We study the Galois group $G_f$ of a random polynomial $f$ of height at most $H$ in the family of polynomials of degree $2n$ satisfying the twisted reciprocal relation $f(x) = x^{2n}/b^n \cdot f(b/x)$, which arise in a wide variety of applications. Our main result is a theorem of van der Waerden-Bhargava type: the probability that $G_f$ is not the full hyperoctahedral group $S_2 \wr S_n$ is $\Theta(H^{-1}\log H)$, independent of $b$, with the leading-order group $G_1$ being of index $2$. This paper is a companion to a recent paper by the authors and Bertelli addressing reciprocal polynomials (i.e. the case $b = 1$).