2025/03/11 by Robin Zhang, Zhang, Robin
Mathematics · #11D25 #11D41 #11D45 (Primary) 05E16 #11D72 #13F60 #14G05 (Secondary) #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2503.08800
openalex publication_date 2025/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We determine the number of positive integral points on n-dimensional affine varieties associated to arbitrary n × 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 E7 and E8 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.