The main principle is that counting roots corresponds to solving start systems. Algorithms to illustrate this principle will be shown for little examples for the three classes of polynomial systems.
For dense polynomial systems, the computation of generalized permanents model the resolution of linear-product start systems. The algorithms to compute mixed volumes lead to polyhedral homotopies to solve sparse polynomial systems. The localization posets describe the structure of the cellation of the Grassmannian used to set up the Pieri deformations.