  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.
