On the one dimensional polynomial, regular and regulous images of closed balls and spheres
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918085550669824 |
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| author | Fernando, José F. |
| author_facet | Fernando, José F. |
| contents | We present a full geometric characterization of the $1$-dimensional (semialgebraic) images $S$ of either $n$-dimensional closed balls $\overline{\mathcal B}_n\subset{\mathbb R}^n$ or $n$-dimensional spheres ${\mathbb S}^n\subset{\mathbb R}^{n+1}$ under polynomial, regular and regulous maps for some $n\geq1$. In all the previous cases one can find an alternative polynomial, regular or regulous map on either $\overline{\mathcal B}_1:=[-1,1]$ or ${\mathbb S}^1$ such that $S$ is the image under such map of either $\overline{\mathcal B}_1:=[-1,1]$ or ${\mathbb S}^1$. As a byproduct, we provide a full characterization of the images of ${\mathbb S}^1\subset{\mathbb C}\equiv{\mathbb R}^2$ under Laurent polynomials $f\in{\mathbb C}[{\tt z},{\tt z}^{-1}]$, taking advantage of some previous works of Kobalev-Yang and Wilmshurst. We also alternatively prove that all polynomial maps ${\mathbb S}^k\to{\mathbb S}^1$ are constant if $k\geq2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09943 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the one dimensional polynomial, regular and regulous images of closed balls and spheres Fernando, José F. Algebraic Geometry Primary: 14P10, 26C05, 26C15, Secondary: 14P05, 14P25, 42A05 We present a full geometric characterization of the $1$-dimensional (semialgebraic) images $S$ of either $n$-dimensional closed balls $\overline{\mathcal B}_n\subset{\mathbb R}^n$ or $n$-dimensional spheres ${\mathbb S}^n\subset{\mathbb R}^{n+1}$ under polynomial, regular and regulous maps for some $n\geq1$. In all the previous cases one can find an alternative polynomial, regular or regulous map on either $\overline{\mathcal B}_1:=[-1,1]$ or ${\mathbb S}^1$ such that $S$ is the image under such map of either $\overline{\mathcal B}_1:=[-1,1]$ or ${\mathbb S}^1$. As a byproduct, we provide a full characterization of the images of ${\mathbb S}^1\subset{\mathbb C}\equiv{\mathbb R}^2$ under Laurent polynomials $f\in{\mathbb C}[{\tt z},{\tt z}^{-1}]$, taking advantage of some previous works of Kobalev-Yang and Wilmshurst. We also alternatively prove that all polynomial maps ${\mathbb S}^k\to{\mathbb S}^1$ are constant if $k\geq2$. |
| title | On the one dimensional polynomial, regular and regulous images of closed balls and spheres |
| topic | Algebraic Geometry Primary: 14P10, 26C05, 26C15, Secondary: 14P05, 14P25, 42A05 |
| url | https://arxiv.org/abs/2406.09943 |