标题： 二元二次型的替换动力学 显示英文标题

标题： Replacement dynamics of binary quadratic forms

摘要： 对于一个取值为$S$的关于$m \geq 1$个变量的函数$f$，我们定义了一个动力过程，在该过程中，输出$f(\overline{v})$在迭代过程的每一步恰好替换输入$\overline{v} \in S^m$的一个条目。 我们的研究关注与此过程相关的周期向量。 我们根据其类型对周期向量进行分层，并表征那些周期向量的确定可归结为单变量多项式动力学的类型。 我们随后限制在域$\mathbb{Q}$上的对角二元二次型$f$的情况，并对所有周期类型直到$5$进行有理周期向量的分类。 这包括两种不来源于单变量情况的类型。 显示英文摘要

摘要： For an $S$-valued function $f$ of $m \geq 1$ variables we define a dynamical process in which the output $f(\overline{v})$ replaces exactly one entry of the input $\overline{v} \in S^m$ at each step in the iterative process. Our study focuses on periodic vectors with respect to this process. We define a stratification of periodic vectors according to their type, and characterize types for which the determination of periodic vectors comes down to dynamics of univariate polynomials. We then restrict to the case of a diagonal binary quadratic form $f$ over $\mathbb{Q}$, and classify rational periodic vectors for all types of period up to $5$. This includes two types which do not arise from the univariate case.