A Polyhedral Homotopy Algorithm for Real Zeros

We design a homotopy continuation algorithm, that is based on Viro’s patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients that satisfy certain concavity conditions, it tracks optimal number of solution paths, and it operates entirely over the reals. In more technical terms, we design an algorithm that correctly counts and finds the real zeros of polynomial systems that are located in the unbounded components of the complement of the underlying A-discriminant amoeba. We provide a detailed exposition of connections between Viro’s patchworking method, convex geometry of A-discriminant amoeba complements, and computational real algebraic geometry.