Galois groups of twisted reciprocal polynomials: a uniform asymptotic

We study the Galois group $G_f$ of a random polynomial $f$ in the family of polynomials of degree $2n$ satisfying the twisted reciprocal relation $f(x)=x^{2n}\mathrm{/}{b}^n\cdot f(b\mathrm{/}x)$. We use a Euclidean height adapted to this relation. Our main result is an asymptotic theorem of van der Waerden–Bhargava type: for fixed $b\mathrm{\ne }0$ and $n\ge 4$, the number of polynomials of height at most $H$ whose Galois group is not the full hyperoctahedral group $S_2\wr S_n$ is an explicit constant times $H^n\log H$, with an error of order $H^n$. We determine the dependence of the leading constant on $b$; the leading-order group $G_1$ is of index $2$. This paper is a sequel to a recent paper by Anderson, Bertelli, and the author addressing reciprocal polynomials (i.e. the case $b=1$).