We study the Escape Problem for discrete-time linear dynamical systems over compact semialgebraic sets. We establish a uniform upper bound on the number of iterations it takes for every orbit of a rational matrix to escape a compact semialgebraic set defined over rational data. Our bound is doubly exponential in the ambient dimension, singly exponential in the degrees of the polynomials used to define the semialgebraic set, and singly exponential in the bitsize of the coefficients of these polynomials and the bitsize of the matrix entries. We show that our bound is tight by providing a matching lower bound.

翻译：我们研究离散时间线性动态系统在紧凑的半成形数组上的“逃脱问题”。我们为理性矩阵的每个轨道所需的迭代数量设定了一个统一的上限,以避开根据合理数据定义的紧凑的半成形数组。我们的结合在环境维度上是双倍指数,在用于定义准成形数组的多元数度中是单倍指数,在这些多成数组和矩阵条目的比特大小中是单倍指数。我们通过提供相匹配的较低约束,显示我们的结合是紧的。