The set of hyperbolic equilibria and of invertible zeros on the unit ball is computable

In this note, we construct an algorithm that, on input of a description of a structurally stable planar dynamical flow $f$ defined on the closed unit disk, outputs the exact number of the (hyperbolic) equilibrium points and their locations with arbitrary accuracy. By arbitrary accuracy it is meant that any accuracy required by the input can be achieved. The algorithm can be further extended to a root-finding algorithm that computes the exact number of zeros as well the location of each zero of a continuously differentiable function $f$ defined on the closed unit ball of $\mathbb{R}^{d}$, provided that the Jacobian of $f$ is invertible at each zero of $f$; moreover, the computation is uniform in $f$.