A Survey on the Complexity of Solving Algebraic Systems

This paper presents a lecture on existing algorithms for solving poly-nomial systems with their complexity analysis from our experiments on the subject. It is based on our studies of the complexity of solving para-metric polynomial systems. It is intended to be useful to two groups of people: those who wish to know what work has been done and those who would like to do work in the field. It contains an extensive bibliography to assist readers in exploring the field in more depth. The paper pro-vides different methods and techniques used for representing solutions of algebraic systems that include Rational Univariate Representations (RUR), Gröbner bases, etc