Decidability of the Orbit Problem Over $\mathbb{Q}(X)$

Abstract

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 $\operatorname{Aut}(F_2)$.