Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains

Abstract : In this paper we consider the computational complexity of solving initial-value problems defined with analytic ordinary differential equations (ODEs) over unbounded domains in $\R^n$ and $\C^n$. We show that the solution can be computed in polynomial time over its maximal interval of definition, provided it satisfies a very generous bound on its growth, and that the function admits an analytic extension over a strip on the complex plane.