A way of solving systems of polynomial equations over $\mathbb{R}$ or $\mathbb{C}$ is using homotopy continuation. Roughly speaking this method uses a homotopy that starts from some system of equations of which the solutions are known and goes to the the system of equations that we want to solve. By numerically tracking the solutions as the system of equations changes into the system we want to solve one can find its solutions.
To me, this method seems useless for solving polynomial systems of equations over finite fields. For starters, I don't think that finite fields (or their extensions) have the topological structure that is needed for the concept of homotopy to make sense.
Is the method of homotopy continuation completely doomed in this context?