Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.15875 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914399778766848 |
|---|---|
| author | Anderson, Theresa C. O'Dorney, Evan M. |
| author_facet | Anderson, Theresa C. O'Dorney, Evan M. |
| contents | 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 $b = 1$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15875 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Galois groups of reciprocal polynomials II: Twisted reciprocal polynomials Anderson, Theresa C. O'Dorney, Evan M. Number Theory 11R32, 11R45, 11C08, 11N35, 20E22 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 $b = 1$). |
| title | Galois groups of reciprocal polynomials II: Twisted reciprocal polynomials |
| topic | Number Theory 11R32, 11R45, 11C08, 11N35, 20E22 |
| url | https://arxiv.org/abs/2603.15875 |