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arXiv · 2603.15875

Galois groups of reciprocal polynomials II: Twisted reciprocal polynomials

Theresa C. Anderson·Evan M. O'Dorney
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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 $Θ(H^{-1}\log H)$, 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 $b = 1$).

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Theresa C. Anderson, Evan M. O'Dorney. 2026-03-16. Galois groups of reciprocal polynomials II: Twisted reciprocal polynomials. https://arxiv.org/abs/2603.15875

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