A stratified polyhedral homotopy method for sampling positive-dimensional zero sets of polynomial systems

Numerical algebraic geometry revolves around the study of solutions to polynomial systems via numerical methods. The polyhedral homotopy of Huber and Sturmfels for computing isolated solutions and the concept of witness sets as numerical representations of non-isolated solution components, put forth by Sommese and Wampler, are two pillars of this field. In this paper, we show that a modified polyhedral homotopy places the computation of isolated solutions and sample sets for non-isolated components into a single, unified framework. In certain cases, this method also leads to a natural decomposition of the BKK bound into a sum of its local contributions.