发表机构
Rose-Hulman Institute of Technology（罗斯-赫尔曼理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了$\boldsymbol{\text{Q}(X)}$上的轨道问题可判定，并利用该结果得出辫群、$\boldsymbol{\text{Aut}(F_2)}$等群的轨道幂问题也可判定。
AI 中文摘要
轨道问题是指：给定$x,y \boldsymbol{\text{Q}}^n$和$n \times n$矩阵$A$，是否存在自然数$i$使得$A^i x = y$。1980年，Kannan和Lipton证明了轨道问题是可判定的。本文证明了轨道问题的一个推广形式也是可判定的，该推广形式中基域为$\boldsymbol{\text{Q}}(X)$，其中$X$是可数的超越数集合。我们针对任意群$G$定义了轨道幂问题：给定$x,y \boldsymbol{\text{G}}$，是否存在整数$n$使得$x^n = y$。随后利用上述主要结果，证明了多种群的轨道幂问题是可判定的，包括辫群和$\boldsymbol{\text{Aut}(F_2)}$。
英文摘要
The orbit problem is the problem of whether, given $x, y\in \mathbb{Q}^n$ and an $n\times n$ matrix $A$, there exists an $i\in\mathbb{N}$ such that $A^ix = y$. In 1980, Kannan and Lipton proved that the orbit problem is decidable. We show that a generalization of the orbit problem, where the field is $\mathbb{Q}(X)$ for $X$ a countable set of transcendentals, is also decidable. We define the orbit power problem for an arbitrary group $G$ to be the problem of when, given $x, y\in G$, there exists an $n\in\mathbb{Z}$ such that $x^n = y$. We then use the main result to show that the power orbit problem is decidable for an assortment of groups, including the braid groups and $\operatorname{Aut}(F_2)$.
Comments7 pages
