  Numerical algebraic geometry revolves around the study of solutions to
polynomial systems via numerical method. 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 of the fundamental tools in this field. In this
paper, we show that a modified polyhedral homotopy can reveal sample sets of
non-isolated solution components, akin to witness sets, as by-products from the
process of computing isolated solutions. In certain cases, this method also
leads to a natural decomposition of the BKK bound into a sum of local
contributions from individual irreducible components.
