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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2109.05985 |
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| _version_ | 1866917840838197248 |
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| author | Frick, Florian Harrison, Michael |
| author_facet | Frick, Florian Harrison, Michael |
| contents | We show that any embedding $\mathbb{R}^d \to \mathbb{R}^{2d+2^{γ(d)}-1}$ inscribes a trapezoid or maps three points to a line, where $2^{γ(d)}$ is the smallest power of $2$ satisfying $2^{γ(d)} \geq ρ(d)$, and $ρ(d)$ denotes the Hurwitz--Radon function. The proof is elementary and includes a novel application of nonsingular bilinear maps. As an application, we recover recent results on the nonexistence of affinely $3$-regular maps, for infinitely many dimensions $d$, without resorting to sophisticated algebraic techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_05985 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On inscribed trapezoids and affinely 3-regular maps Frick, Florian Harrison, Michael Geometric Topology 53A07, 57R40 We show that any embedding $\mathbb{R}^d \to \mathbb{R}^{2d+2^{γ(d)}-1}$ inscribes a trapezoid or maps three points to a line, where $2^{γ(d)}$ is the smallest power of $2$ satisfying $2^{γ(d)} \geq ρ(d)$, and $ρ(d)$ denotes the Hurwitz--Radon function. The proof is elementary and includes a novel application of nonsingular bilinear maps. As an application, we recover recent results on the nonexistence of affinely $3$-regular maps, for infinitely many dimensions $d$, without resorting to sophisticated algebraic techniques. |
| title | On inscribed trapezoids and affinely 3-regular maps |
| topic | Geometric Topology 53A07, 57R40 |
| url | https://arxiv.org/abs/2109.05985 |