Solving polynomial equations: Characteristic sets and triangular systems

We review, compare and experiment with two methods for solving systems of polynomial equations. One of the methods is developed by Wu based on Ritt's concept of characteristic sets, and the other is proposed by the author in extending the idea of Seidenberg's elimination theory. We present the two methods in parallel and explain how a system of polynomial equations can be solved by computing the medial set, characteristic set, characteristic (irreducible) series, principal triangular system and (irreducible) triangular series of the corresponding polynomial system. Experimental data on 50 examples are provided which demonstrate the applicability and efficiency of the methods.