After touching on prerequisite knowledge and introducing the field of computational algebraic geometry, we invent and implement a new algorithm to find solutions of generic systems of multivariate polynomials. The novel approach we employ is analytically continuing through a random set of points in an extended space—this addresses the main shortcoming of previous solvers, which is numerical instability of path trackers near branch points. We rigorously justify the correctness and completeness of our algorithm by drawing from analytic continuation, generalized Maclaurin series (Pade techniques), and the action of the fundamental group. We then demonstrate our algorithm’s competitiveness by testing it on benchmark systems and comparing its performance to state-of-the-art solvers.