Complexity of sparse polynomial solving 2: renormalization

Abstract Renormalized homotopy continuation on toric varieties is introduced as a tool for solving sparse systems of polynomial equations or exponential sums. The cost depends renormalized condition length, defined line integral the number along all lifted paths. theory developed in this paper leads to algorithm tracking solutions between two generic with same structure. randomized, sense that it follows random path systems. probability success one. In order produce an expected bound, several invariants depending solely supports are introduced. For instance, mixed area quermassintegral generalizes surface way volume ordinary volume. facet gap measures each 1-cone fan and support polytope, how close supporting hyperplane nearest vertex. Once fixed, input coefficients through invariants: imbalance absolute values coefficients. This nonuniform complexity bound terms those invariants.