Symmetries of various sets of polynomials
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911082629562368 |
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| author | Seguin, Béranger |
| author_facet | Seguin, Béranger |
| contents | Let $K$ be a field of characteristic $0$, and let $k \geq 2$ be an integer. We prove that every $K$-linear bijection $f \colon K[X] \to K[X]$ 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., that there are elements $a, c \in K^{\times}$ and $b \in K$ 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 $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09118 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetries of various sets of polynomials Seguin, Béranger Commutative Algebra Number Theory 47B49, 15A86, 12E05, 26C10, 12E25 Let $K$ be a field of characteristic $0$, and let $k \geq 2$ be an integer. We prove that every $K$-linear bijection $f \colon K[X] \to K[X]$ 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., that there are elements $a, c \in K^{\times}$ and $b \in K$ 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 $K$. |
| title | Symmetries of various sets of polynomials |
| topic | Commutative Algebra Number Theory 47B49, 15A86, 12E05, 26C10, 12E25 |
| url | https://arxiv.org/abs/2407.09118 |