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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.10870 |
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| _version_ | 1866911383597088768 |
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| author | Apolonskaya, Ksenia Musin, Oleg R. |
| author_facet | Apolonskaya, Ksenia Musin, Oleg R. |
| contents | This paper studies the minimal number of vertices $λ(n,d)$ required in a triangulation of the $n$-sphere to admit a simplicial map to the boundary of a $(n+1)$-simplex with a given degree $d$. We establish upper bounds for $λ(n,d)$ in dimensions $n \geq 3$. Furthermore, we provide exact formulas for small values of $d$, showing that $λ(n,d)=n+d+3$ for $n \geq 3$ and $d=2,3,4$. A key technical result is the identity $λ(n,d) = λ(d-1,d) + n - d + 1$ for $n \geq d$, which allows us to reduce higher-dimensional cases to lower-dimensional ones. The proofs involve constructive methods based on local modifications of triangulations and combinatorial arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10870 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimal simplicial spherical mappings with a given degree Apolonskaya, Ksenia Musin, Oleg R. Combinatorics This paper studies the minimal number of vertices $λ(n,d)$ required in a triangulation of the $n$-sphere to admit a simplicial map to the boundary of a $(n+1)$-simplex with a given degree $d$. We establish upper bounds for $λ(n,d)$ in dimensions $n \geq 3$. Furthermore, we provide exact formulas for small values of $d$, showing that $λ(n,d)=n+d+3$ for $n \geq 3$ and $d=2,3,4$. A key technical result is the identity $λ(n,d) = λ(d-1,d) + n - d + 1$ for $n \geq d$, which allows us to reduce higher-dimensional cases to lower-dimensional ones. The proofs involve constructive methods based on local modifications of triangulations and combinatorial arguments. |
| title | Minimal simplicial spherical mappings with a given degree |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.10870 |