Bounding the Escape Time of a Linear Dynamical System over a Compact Semialgebraic Set

TL;DR

This paper establishes a tight, doubly exponential upper bound on the number of iterations for linear dynamical systems to escape a compact semialgebraic set, considering the system's algebraic complexity.

Contribution

It provides the first explicit uniform upper bound on escape time for such systems, matching a new lower bound, with bounds expressed in algebraic and geometric parameters.

Findings

Upper bound is doubly exponential in dimension

Bound is tight, matching a lower bound

Results depend on polynomial degrees and coefficient bitsize

Abstract

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.