Empty simplices of large width
An empty simplex is a lattice simplex in which vertices are the only lattice points. We show two constructions leading to the first known empty simplices of width larger than their dimension: ◦ We introduce cyclotomic simplices and exhaustively compute all the cyclotomic simplices of dimension...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001312/type/journal_article |
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| Summary: | An empty simplex is a lattice simplex in which vertices are the only lattice points. We show two constructions leading to the first known empty simplices of width larger than their dimension:
◦
We introduce cyclotomic simplices and exhaustively compute all the cyclotomic simplices of dimension
$10$
and volume up to
$2^{31}$
. Among them, we find five empty ones of width
$11$
and none of larger width.
◦
Using circulant matrices of a very specific form, we construct empty simplices of arbitrary dimension d and width growing asymptotically as
$d/\operatorname {\mathrm {arcsinh}}(1) \sim 1.1346\,d$
.
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| ISSN: | 2050-5094 |