article Beranger Fabrice Seguin author 102487 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 $K$. 2025 eng 2407.0911810.1007/s13366-025-00800-2 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> @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} } 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>. 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>. B.F. Seguin, Beiträge Zur Algebra Und Geometrie (2025). 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> 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>. 581872025-01-15T11:25:18Z2025-07-16T13:51:54Z