Squared edge lengths of regular simplices with rational vertices
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917486656487424 |
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| author | Kominers, Scott Duke |
| author_facet | Kominers, Scott Duke |
| contents | We determine exactly which positive rational numbers occur as squared edge lengths of regular $d$-simplices with vertices in $\mathbb{Q}^n$. The answer exhibits a sharp stabilization phenomenon: once $n-d\geq 3$, every positive rational number occurs, while codimensions $0$, $1$, and $2$ are governed by explicit square-class, norm-group, and Hilbert-symbol conditions. The proof reduces simplex realizability to the Hasse--Minkowski classification of rational quadratic forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12268 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Squared edge lengths of regular simplices with rational vertices Kominers, Scott Duke Number Theory Metric Geometry Primary 11E12, 51M20, Secondary 11E81, 51N20 We determine exactly which positive rational numbers occur as squared edge lengths of regular $d$-simplices with vertices in $\mathbb{Q}^n$. The answer exhibits a sharp stabilization phenomenon: once $n-d\geq 3$, every positive rational number occurs, while codimensions $0$, $1$, and $2$ are governed by explicit square-class, norm-group, and Hilbert-symbol conditions. The proof reduces simplex realizability to the Hasse--Minkowski classification of rational quadratic forms. |
| title | Squared edge lengths of regular simplices with rational vertices |
| topic | Number Theory Metric Geometry Primary 11E12, 51M20, Secondary 11E81, 51N20 |
| url | https://arxiv.org/abs/2605.12268 |