Counting in Vieta graphs over ${\mathbb{F}}_p$

Abstract

We introduce and study a finite simple graph of algebraic origin: the Vieta graph on the solution set over ${\mathbb{F}}_p$ to a symmetric, multivariate equation which is quadratic in each variable. This construction is a broad generalization of the Markoff graph over ${\mathbb{F}}_p$, 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.

47 pages, 9 figures. Comments are welcome