The higher-dimensional version of Kannan and Lipton's Orbit Problem asks whether it is decidable if a target vector space can be reached from a starting point under repeated application of a linear transformation. This problem has remained open since its formulation, and in fact generalizes Skolem's Problem -- a long-standing open problem concerning the existence of zeros in linear recurrence sequences. Both problems have traditionally been studied using algebraic and number theoretic machinery. In contrast, this paper reduces the Orbit Problem to an equivalent version in real projective space, introducing a basic geometric reference for examining and deciding problem instances. We find this geometric toolkit enables basic proofs of sweeping assertions concerning the decidability of certain problem classes, including results where the only other known proofs rely on sophisticated number-theoretic arguments.

翻译：Kannan与Lipton轨道问题的高维推广形式，探究在给定线性变换的重复作用下，目标向量空间能否从初始点可达。该问题自提出以来悬而未决，实际上它推广了Skolem问题——一个关于线性递推序列零点存在性的长期未解难题。传统上这两个问题均借助代数与数论工具进行研究。与之相对，本文将轨道问题约化为实射影空间中的等价形式，引入了用于检验与判定问题实例的基本几何参照系。我们发现这一几何工具集能够为特定问题类可判定性的广泛论断提供基础性证明，其中部分结论此前仅能通过复杂的数论论证得以验证。