Counting in Vieta graphs over \mathbbF_p

2026/08/04 by Bogdan Nica
Mathematics · #math.NT #math.CO #msc:11D25 #msc:05C25 #msc:11L10

paper · pdf

47 pages, 9 figures. Comments are welcome

arxiv created 2026/08/04 · arxiv updated 2026/08/05

Abstract

We introduce and study a finite simple graph of algebraic origin: the Vieta graph on the solution set over \mathbbF_p to a symmetric, multivariate equation which is quadratic in each variable. This construction is a broad generalization of the Markoff graph over \mathbbF_p, extensively studied in the recent literature. We give a systematic approach, partly based on quadratic character sums, to the following basic counting questions: how many vertices does a Vieta graph have, and what is the degree distribution? We focus on explicit counts, addressing the low-dimensional cases in three and four variables.

Citations

• Topological properties of generalized Markoff mod p graphs
• Orderings of k-Markov Numbers
• Divisibility by p for Markoff-like Surfaces
• A new proof of Chen's theorem for Markoff graphs
• SL(2,Z)-matrixizations of generalized Markov numbers
• Uniqueness theorem of generalized Markov numbers that are prime powers
• Connectivity of Markoff mod-p graphs and maximal divisors
• Orbits on K3 Surfaces of Markoff Type
• Generalization of Markov Diophantine equation via generalized cluster algebra
• Non-planarity of Markoff graphs mod p
• Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlevé 6
• Nonabelian level structures, Nielsen equivalence, and Markoff triples
• Experiments with the Markoff surface
• Markoff Triples and Strong Approximation
• Trace Coordinates on Fricke spaces of some simple hyperbolic surfaces