{"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/quasi-period-collapse-in-half-integral","title":"Quasi-period collapse in half-integral polygons","arxiv_id":"2405.13404","date":"2024-05-22","proceeding":null,"authors":["Martin Bohnert"],"abstract":"A half-integral polygon with quasi-period collapse behaves similarly to a lattice polygon in the sense that the number of lattice points in its integer dilates can be calculated as values of a polynomial, its Ehrhart polynomial. As a main result, we classify the Ehrhart polynomials of all half-integral non-lattice polygons with quasi-period collapse. In particular, we obtain that for any positive integer $i$, the polynomial $\\frac{4i+5}{2}t^2+\\frac{2i+7}{2}t+1\\in \\mathbb{Q}[t]$ is an Ehrhart polynomial of a rational polygon, which was an open question for $i>1$. We also study some extreme cases in detail. In particular, we show that up to affine unimodular equivalence there exist exactly $30$ half-integral non-lattice polygons with quasi-periodic collapse with exactly one interior lattice point, which are the dual polygons of the $30$ LDP polygons of Gorenstein index $2$. Furthermore, we classify all half-integral polygons with quasi-period collapse with at most $6$ interior lattice points or with $i\\geq 1$ interior lattice points and the maximum possible number $2i+7$ of boundary lattice points.","url_abs":"https://arxiv.org/abs/2405.13404v1","url_pdf":"https://arxiv.org/pdf/2405.13404v1.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":"quasi-period-collapse-in-half-integral","repo_url":"https://github.com/mbohnert/period_collapse_half_integral","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}