Abstract:

Central configurations are solutions of parametric algebraic systems. Elimination methods are techniques of solving polynomial systems by eliminating all but one variable, and then extending partial solutions to those of the system. Various elimination methods will be discussed in this talk.

First, I will review the spirit of elimination methods by resultants. One of the most impressive property about them is how they “preserve and accumulate” intersection multiplicities of isolated zeros. However, such tool becomes impractical when there are more than two equations. Moreover, computing the determinants of large matrix with symbolic entries becomes easily infeasible. Therefore, Groebner basis may be a better tool for systems of more than two equations. However, such tool may not be used in studying bifurcations. Also, considering working with parameters, it may take some tricks to take care of the “specialization” issues when it is impractical to compute a comprehensive Groebner basis. I will present some of them. Finally, by a procedure of “Euclidean algorithm,” we introduce the elimination method by Wu’s pseudodivision.