On the Subspace Orbit Problem and the Simultaneous Skolem Problem

TL;DR

This paper investigates the decidability of the Orbit Problem for subspace targets of varying dimensions, establishing decidability for logarithmic dimensions and equivalence to the Skolem Problem for linear dimensions.

Contribution

It proves the Orbit Problem is decidable for subspaces with logarithmic dimension and shows linear dimension cases are as hard as the Skolem Problem.

Findings

Decidability for subspaces with logarithmic dimension in the orbit dimension.

NP^RP complexity bound for the logarithmic dimension case.

Equivalence of the linear dimension case to the Skolem Problem.

Abstract

The Orbit Problem asks whether the orbit of a point under a matrix reaches a given target set. When the target is a single point, the problem was shown to be decidable in polynomial time by Kannan and Lipton. This decidability result was later extended by Chonev et al. to targets of dimension 3 (in arbitrary ambient dimension), but decidability remains open for subspaces of dimension 4. At the other extreme, the special case of the Orbit Problem in which the target set is a hyperplane of codimension 1 is equivalent to the Skolem Problem for linear recurrence sequences, whose decidability has been open for many decades. In this paper, we show that the Orbit Problem is decidable if the target subspace has dimension logarithmic in the dimension of the orbit. Over rationals, we moreover obtain a complexity bound NP^RP in this case. On the other hand, we show that the version of the Orbit Problem where the dimension of the target subspace is linear in the dimension of the orbit is as hard as the Skolem Problem.