Solving Polynomial Systems With Tropical Methods

In this thesis, we develop a polyhedral method to solve polynomial systems. We are primarily interested in obtaining the Puiseux series representations of positive dimensional solution sets for square polynomial systems and systems, which consist of more equations than unknowns. By developing our polyhedral method, we aim to generalize polyhedral homotopies. Our polyhedral method can be seen as the symbolic-numeric version of the fundamental theorem of tropical algebraic geometry. We illustrate our polyhedral method on the cyclic n-roots problems and offer a tropical perspective on the lemma of Backelin.