Symmetries of various sets of polynomials

B.F. Seguin, Beiträge Zur Algebra Und Geometrie (2025).

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Journal Article | English
Abstract
Let $K$ be a field of characteristic $0$ and $k \geq 2$ be an integer. We prove that every $K$-linear bijection $f : K[X] \to K[X]$ strongly preserving the set of $k$-free polynomials (or the set of polynomials with a $k$-fold root in $K$) is a constant multiple of a $K$-algebra automorphism of $K[X]$, i.e., there are elements $a, c \in K^{\times}$, $b \in K$ such that $f(P)(X) = c P(a X + b)$. When $K$ is a number field or $K=\mathbb{R}$, we prove that similar statements hold when $f$ preserves the set of polynomials with a root in $K$.
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Beiträge zur Algebra und Geometrie
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Seguin BF. Symmetries of various sets of polynomials. Beiträge zur Algebra und Geometrie. Published online 2025. doi:10.1007/s13366-025-00800-2
Seguin, B. F. (2025). Symmetries of various sets of polynomials. Beiträge Zur Algebra Und Geometrie. https://doi.org/10.1007/s13366-025-00800-2
@article{Seguin_2025, title={Symmetries of various sets of polynomials}, DOI={10.1007/s13366-025-00800-2}, journal={Beiträge zur Algebra und Geometrie}, author={Seguin, Beranger Fabrice}, year={2025} }
Seguin, Beranger Fabrice. “Symmetries of Various Sets of Polynomials.” Beiträge Zur Algebra Und Geometrie, 2025. https://doi.org/10.1007/s13366-025-00800-2.
B. F. Seguin, “Symmetries of various sets of polynomials,” Beiträge zur Algebra und Geometrie, 2025, doi: 10.1007/s13366-025-00800-2.
Seguin, Beranger Fabrice. “Symmetries of Various Sets of Polynomials.” Beiträge Zur Algebra Und Geometrie, 2025, doi:10.1007/s13366-025-00800-2.

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arXiv 2407.09118

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