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A positive Siegel theorem: Dynkin friezes and positive Mordell-Schinzel
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A positive Siegel theorem: Dynkin friezes and positive Mordell-Schinzel
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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.
Forward citations
Cited by 2 Pith papers
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On maximal Dynkin friezes
Explicit constructions on affine cluster varieties produce B_n and D_n Dynkin friezes over positive integers with largest entries F_{n+1}F_{n+2}-1 and F_n F_{n+1}-1 respectively, conjectured to be maximal.
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Frieze patterns in representation theory
A survey of results linking frieze patterns to polygon triangulations, Grassmannian cluster algebras, and Grassmannian cluster categories, with focus on recent links to cluster categories.
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