{"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/enumerating-polytropes","title":"Enumerating Polytropes","arxiv_id":"1310.2012","date":"2013-10-08","proceeding":null,"authors":["Ngoc Mai Tran"],"abstract":"Polytropes are both ordinary and tropical polytopes. We show that tropical types of polytropes in $\\mathbb{TP}^{n-1}$ are in bijection with cones of a certain Gr\\\"{o}bner fan $\\mathcal{GF}_n$ in $\\mathbb{R}^{n^2 - n}$ restricted to a small cone called the polytrope region. These in turn are indexed by compatible sets of bipartite and triangle binomials. Geometrically, on the polytrope region, $\\mathcal{GF}_n$ is the refinement of two fans: the fan of linearity of the polytrope map appeared in \\cite{tran.combi}, and the bipartite binomial fan. This gives two algorithms for enumerating tropical types of polytropes: one via a general Gr\\\"obner fan software such as \\textsf{gfan}, and another via checking compatibility of systems of bipartite and triangle binomials. We use these algorithms to compute types of full-dimensional polytropes for $n = 4$, and maximal polytropes for $n = 5$.","url_abs":"http://arxiv.org/abs/1310.2012v4","url_pdf":"http://arxiv.org/pdf/1310.2012v4.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":"enumerating-polytropes","repo_url":"https://github.com/princengoc/polytropes","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}