  In this paper we prove that computing the solution of an initial-value
problem $\dot{y}=p(y)$ with initial condition $y(t_0)=y_0\in\R^d$ at time
$t_0+T$ with precision $e^{-\mu}$ where $p$ is a vector of polynomials can be
done in time polynomial in the value of $T$, $\mu$ and $Y=\sup_{t_0\leqslant
u\leqslant T}\infnorm{y(u)}$. Contrary to existing results, our algorithm works
for any vector of polynomials $p$ over any bounded or unbounded domain and has
a guaranteed complexity and precision. In particular we do not assume $p$ to be
fixed, nor the solution to lie in a compact domain, nor we assume that $p$ has
a Lipschitz constant.
