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Galois groups of twisted reciprocal polynomials: a uniform asymptotic
Abstract
We study the Galois group Gf of a random polynomial f in the family of polynomials of degree 2n satisfying the twisted reciprocal relation f(x) = x2n/bn · f(b/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 ≠ 0 and n ≥ 4, the number of polynomials of height at most H whose Galois group is not the full hyperoctahedral group S2 wr Sn is an explicit constant times Hnlog H, with an error of order Hn. We determine the dependence of the leading constant on b; the leading-order group G1 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).
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BibTeX
@misc{odorney2026galois,
title = {{Galois groups of twisted reciprocal polynomials: a uniform asymptotic}},
author = {Evan M. O'Dorney},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31613},
eprint = {2609.31613},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31613},
}
RIS
TY - UNPB
TI - Galois groups of twisted reciprocal polynomials: a uniform asymptotic
AU - Evan M. O'Dorney
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31613
UR - https://arxiv.org/abs/2609.31613
AB - 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/b^n · f(b/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 ≠ 0 and n ≥ 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^nlog 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).
ER -