Document Type
Article
Publication Date
8-2026
Abstract
The orbit problem is the problem of whether, given x, y ∈ Qn and an n × n matrix A, there exists an i ∈ N such that Aix = 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 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 ∈ G, there exists an n ∈ Z such that xn = 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 Aut(F2).
Recommended Citation
Holden, Joshua and Renner, Alexa, "Decidability of the Orbit Problem Over Q(X)" (2026). Mathematical Sciences Technical Reports (MSTR). 189.
https://scholar.rose-hulman.edu/math_mstr/189