Polynomial systems appear in many different fields of study. Many important problems can be reduced to solving systems of polynomial equations and usually the coefficients involve parameters. This thesis is devoted to finding practical ways to solve such problems from two fields, the studies of central configurations from the Newtonian N-body problem and Maxwell's conjecture about the electric potential created by point charges.
Central configurations play an important role in the study of celestial mechanics.
They determine some special solutions of the Newton's laws of motion and lead to
explicit expression of the solutions. After some changes of the coordinates, we can
describe the central configurations as zeros of a system of polynomials, where the
coefficients of each polynomial are polynomials in the masses. Therefore, the problem of counting central configurations becomes counting the positive zeros of parametric polynomial systems.
A problem studied by James C. Maxwell back in the 19th century is about finding
an upper bound of the number of non-degenerate equilibrium points of the electric potential created by point charges. In the case of 3 point charges, he conjectured that there are at most 4 such equilibrium points. After given proper coordinates, the problem also becomes to count positive zeros of a parametric polynomial system. In Chapter 1, we will introduce these two problems and derive some parametric polynomial systems for which we will count positive zeros in Chapter 4. Some open questions from these two fields of studies will be given in Chapter 5.
Our methods of counting positive zeros are based on classic tools such as resultants, subresultant sequences, and Hermite quadratic forms. Recently developed tools like Groebner bases make it possible to let computers perform symbolic computations of polynomials and count zeros by applying classic results. A computer algebra system (CAS), for example Mathematica, is the software to do such computations. In Chapter 2, we present those tools and demonstrate how to count zeros for polynomial systems with real or complex coefficients in a CAS.
When it comes to counting zeros of parametric polynomial systems, we want to
count zeros of all the real polynomial systems obtained by substituting real numbers for parameters. For example, when there is one parameter, we may want to know the numbers of positive zeros for real polynomial systems obtained by substituting parameters in an open interval (a, b). When there are two parameters, we may want to count positive zeros for all real polynomial systems obtained by substituting parameters with real pairs in an open region in R2. Our main contributions in this thesis are finding methods to achieve that goal based on standard computer algebra tools and applying these methods to some enumeration problems of central configurations and some special cases of Maxwell's conjecture. We will outline our methods and develop sufficient tools in Chapter 3.