{"year":"2024","publication":"arXiv:2407.09118","_id":"58187","external_id":{"arxiv":["2407.09118"]},"date_updated":"2025-01-15T11:35:35Z","type":"preprint","title":"Symmetries of various sets of polynomials","author":[{"first_name":"Beranger Fabrice","last_name":"Seguin","full_name":"Seguin, Beranger Fabrice","id":"102487"}],"date_created":"2025-01-15T11:25:18Z","status":"public","citation":{"ieee":"B. F. Seguin, “Symmetries of various sets of polynomials,” <i>arXiv:2407.09118</i>. 2024.","ama":"Seguin BF. Symmetries of various sets of polynomials. <i>arXiv:240709118</i>. Published online 2024.","mla":"Seguin, Beranger Fabrice. “Symmetries of Various Sets of Polynomials.” <i>ArXiv:2407.09118</i>, 2024.","short":"B.F. Seguin, ArXiv:2407.09118 (2024).","bibtex":"@article{Seguin_2024, title={Symmetries of various sets of polynomials}, journal={arXiv:2407.09118}, author={Seguin, Beranger Fabrice}, year={2024} }","chicago":"Seguin, Beranger Fabrice. “Symmetries of Various Sets of Polynomials.” <i>ArXiv:2407.09118</i>, 2024.","apa":"Seguin, B. F. (2024). Symmetries of various sets of polynomials. In <i>arXiv:2407.09118</i>."},"user_id":"102487","language":[{"iso":"eng"}],"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 $K$."}]}