AI 中文总结
该研究引入有限域F_p上的Vieta图（Markoff图的推广），基于二次特征和系统方法研究其顶点数与度分布等计数问题，处理了三、四变量的低维显式计数情形。
AI 中文摘要
我们引入并研究一种源自代数的有限简单图：有限域F_p上对每个变量均为二次的对称多元方程解集合上的Vieta图，该构造是近年文献中广泛研究的有限域F_p上的Markoff图的广泛推广。我们给出一种部分基于二次特征和的系统方法，以解决以下基本计数问题：Vieta图有多少个顶点，其度分布如何？我们聚焦于显式计数，处理三变量和四变量的低维情形。
英文摘要
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.
Comments47 pages, 9 figures. Comments are welcome
