{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-cyclicity-rank-of-empty-lattice-simplices","title":"The cyclicity rank of empty lattice simplices","arxiv_id":"2407.01179","date":"2024-07-01","proceeding":null,"authors":["Lukas Abend","Matthias Schymura"],"abstract":"We are interested in algebraic properties of empty lattice simplices $\\Delta$, that is, $d$-dimensional lattice polytopes containing exactly $d+1$ points of the integer lattice $\\mathbb{Z}^d$. The cyclicity rank of $\\Delta$ is the minimal number of cyclic subgroups that the quotient group of $\\Delta$ splits into. It is known that up to dimension $d \\leq 4$, every empty lattice $d$-simplex is cyclic, meaning that its cyclicity rank is at most $1$. We determine the maximal possible cyclicity rank of an empty lattice $d$-simplex for dimensions $d \\leq 8$, and determine the asymptotics of this number up to a logarithmic term.","url_abs":"https://arxiv.org/abs/2407.01179v2","url_pdf":"https://arxiv.org/pdf/2407.01179v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-cyclicity-rank-of-empty-lattice-simplices","repo_url":"https://github.com/mschymura/cyclicity-rank-computations","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}