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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| Format: | Article |
| Language: | English |
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001312/type/journal_article |
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| author | Joseph Doolittle Lukas Katthän Benjamin Nill Francisco Santos |
| author_facet | Joseph Doolittle Lukas Katthän Benjamin Nill Francisco Santos |
| author_sort | Joseph Doolittle |
| collection | DOAJ |
| description | 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
$d/\operatorname {\mathrm {arcsinh}}(1) \sim 1.1346\,d$
.
|
| format | Article |
| id | doaj-art-c9daca6d3d8b4d9e9b262f0c7186bcc4 |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-c9daca6d3d8b4d9e9b262f0c7186bcc42025-02-03T10:39:43ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.131Empty simplices of large widthJoseph Doolittle0Lukas Katthän1https://orcid.org/0000-0002-1456-0514Benjamin Nill2https://orcid.org/0000-0001-5671-8271Francisco Santos3https://orcid.org/0000-0003-2120-9068Institut für Geometrie, TU Graz, Graz, Austria; E-mail:Institut für Mathematik, Goethe-Universität Frankfurt, Graz, Germany; E-mail:Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, Germany; E-mail:Dep. of Mathematics, Statistics and Comp. Sci., Univ. of Cantabria, Santander, SpainAn 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 $d/\operatorname {\mathrm {arcsinh}}(1) \sim 1.1346\,d$ . https://www.cambridge.org/core/product/identifier/S2050509424001312/type/journal_article52B2052C0752B15 |
| spellingShingle | Joseph Doolittle Lukas Katthän Benjamin Nill Francisco Santos Empty simplices of large width Forum of Mathematics, Sigma 52B20 52C07 52B15 |
| title | Empty simplices of large width |
| title_full | Empty simplices of large width |
| title_fullStr | Empty simplices of large width |
| title_full_unstemmed | Empty simplices of large width |
| title_short | Empty simplices of large width |
| title_sort | empty simplices of large width |
| topic | 52B20 52C07 52B15 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424001312/type/journal_article |
| work_keys_str_mv | AT josephdoolittle emptysimplicesoflargewidth AT lukaskatthan emptysimplicesoflargewidth AT benjaminnill emptysimplicesoflargewidth AT franciscosantos emptysimplicesoflargewidth |