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Numerical algebraic geometry revolves around the study of solutions topolynomial systems via numerical method. The polyhedral homotopy of Huber andSturmfels for computing isolated solutions and the concept of witness sets asnumerical representations of non-isolated solution components, put forth bySommese and Wampler, are two of the fundamental tools in this field. In thispaper, we show that a modified polyhedral homotopy can reveal sample sets ofnon-isolated solution components, akin to witness sets, as by-products from theprocess of computing isolated solutions. In certain cases, this method alsoleads to a natural decomposition of the BKK bound into a sum of localcontributions from individual irreducible components.

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