---
_id: '58187'
abstract:
- lang: eng
  text: "Let $K$ be a field of characteristic $0$ and $k \\geq 2$ be an integer. We\r\nprove
    that every $K$-linear bijection $f : K[X] \\to K[X]$ strongly preserving\r\nthe
    set of $k$-free polynomials (or the set of polynomials with a $k$-fold root\r\nin
    $K$) is a constant multiple of a $K$-algebra automorphism of $K[X]$, i.e.,\r\nthere
    are elements $a, c \\in K^{\\times}$, $b \\in K$ such that $f(P)(X) = c P(a\r\nX
    + b)$. When $K$ is a number field or $K=\\mathbb{R}$, we prove that similar\r\nstatements
    hold when $f$ preserves the set of polynomials with a root in $K$."
author:
- first_name: Beranger Fabrice
  full_name: Seguin, Beranger Fabrice
  id: '102487'
  last_name: Seguin
citation:
  ama: Seguin BF. Symmetries of various sets of polynomials. <i>Beiträge zur Algebra
    und Geometrie</i>. Published online 2025. doi:<a href="https://doi.org/10.1007/s13366-025-00800-2">10.1007/s13366-025-00800-2</a>
  apa: Seguin, B. F. (2025). Symmetries of various sets of polynomials. <i>Beiträge
    Zur Algebra Und Geometrie</i>. <a href="https://doi.org/10.1007/s13366-025-00800-2">https://doi.org/10.1007/s13366-025-00800-2</a>
  bibtex: '@article{Seguin_2025, title={Symmetries of various sets of polynomials},
    DOI={<a href="https://doi.org/10.1007/s13366-025-00800-2">10.1007/s13366-025-00800-2</a>},
    journal={Beiträge zur Algebra und Geometrie}, author={Seguin, Beranger Fabrice},
    year={2025} }'
  chicago: Seguin, Beranger Fabrice. “Symmetries of Various Sets of Polynomials.”
    <i>Beiträge Zur Algebra Und Geometrie</i>, 2025. <a href="https://doi.org/10.1007/s13366-025-00800-2">https://doi.org/10.1007/s13366-025-00800-2</a>.
  ieee: 'B. F. Seguin, “Symmetries of various sets of polynomials,” <i>Beiträge zur
    Algebra und Geometrie</i>, 2025, doi: <a href="https://doi.org/10.1007/s13366-025-00800-2">10.1007/s13366-025-00800-2</a>.'
  mla: Seguin, Beranger Fabrice. “Symmetries of Various Sets of Polynomials.” <i>Beiträge
    Zur Algebra Und Geometrie</i>, 2025, doi:<a href="https://doi.org/10.1007/s13366-025-00800-2">10.1007/s13366-025-00800-2</a>.
  short: B.F. Seguin, Beiträge Zur Algebra Und Geometrie (2025).
date_created: 2025-01-15T11:25:18Z
date_updated: 2025-07-16T13:51:54Z
doi: 10.1007/s13366-025-00800-2
external_id:
  arxiv:
  - '2407.09118'
language:
- iso: eng
publication: Beiträge zur Algebra und Geometrie
status: public
title: Symmetries of various sets of polynomials
type: journal_article
user_id: '102487'
year: '2025'
...