A positive Siegel theorem: Dynkin friezes and positive Mordell-Schinzel
Robin Zhang
TL;DR
The paper establishes a positive Siegel-type finiteness theorem for n-dimensional affine varieties X_C arising from generalized Cartan matrices, showing finiteness of X_C(ℕ) exactly when t_C > 0 and providing explicit, effective bounds. It builds a bridge between cluster algebra theory and Diophantine geometry by linking positive integral friezes to X_C(ℕ) via frieze polynomials, and resolves the Fontaine–Plamondon conjecture by counting positive integral friezes of type E7 (4400) and E8 (26952). The authors extend Mordell–Schinzel-type results to higher dimensions, exhibiting infinite families of positive integral solutions for Diophantine equations of the form x y z = G(x,y) and their higher-dimensional analogues when the associated Cartan data are infinite type. They use hyperplane reductions, explicit height bounds, and orbit-by-orbit enumeration to achieve both finite-type enumerations and infinite-type Diophantine consequences, with concrete results across classical and exceptional Dynkin types. The work has implications for arithmetic geometry, representation theory, and combinatorics via the explicit correspondence between friezes and cluster-algebra-type varieties.
Abstract
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.