Symmetries of various sets of polynomials Seguin, Beranger Fabrice 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 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_6501 https://ris.uni-paderborn.de/record/58187 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> eng info:eu-repo/semantics/altIdentifier/doi/10.1007/s13366-025-00800-2 info:eu-repo/semantics/altIdentifier/arxiv/2407.09118 info:eu-repo/semantics/closedAccess