A positive Siegel theorem: Dynkin friezes and positive Mordell-Schinzel
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912352411058176 |
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| author | Zhang, Robin |
| author_facet | Zhang, Robin |
| contents | We determine the number of positive integral points on $n$-dimensional affine varieties associated to arbitrary $n \times n$ generalized Cartan matrices. An application to the theory of cluster algebras and combinatorics is the resolution of the Fontaine-Plamondon conjecture, which says that there are exactly $4400$ and $26952$ positive integral friezes of type $E_7$ and $E_8$ respectively. An application to number theory refines and generalizes theorems of Mohanty, Mordell, and Schinzel to the positive integers and higher dimensions by exhibiting examples of Diophantine equations $xyz = G(x, y)$ and $xyzw = G(x, y, z)$ of every degree greater than $3$ with infinitely many positive integral solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08800 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A positive Siegel theorem: Dynkin friezes and positive Mordell-Schinzel Zhang, Robin Number Theory Combinatorics Rings and Algebras 11D45 (Primary) 05E16, 11D25, 11D41, 11D72, 13F60, 14G05 (Secondary) We determine the number of positive integral points on $n$-dimensional affine varieties associated to arbitrary $n \times n$ generalized Cartan matrices. An application to the theory of cluster algebras and combinatorics is the resolution of the Fontaine-Plamondon conjecture, which says that there are exactly $4400$ and $26952$ positive integral friezes of type $E_7$ and $E_8$ respectively. An application to number theory refines and generalizes theorems of Mohanty, Mordell, and Schinzel to the positive integers and higher dimensions by exhibiting examples of Diophantine equations $xyz = G(x, y)$ and $xyzw = G(x, y, z)$ of every degree greater than $3$ with infinitely many positive integral solutions. |
| title | A positive Siegel theorem: Dynkin friezes and positive Mordell-Schinzel |
| topic | Number Theory Combinatorics Rings and Algebras 11D45 (Primary) 05E16, 11D25, 11D41, 11D72, 13F60, 14G05 (Secondary) |
| url | https://arxiv.org/abs/2503.08800 |