Abstract
Numerical algebraic geometry is a collection of methods, based primarily on numerical homotopy continuation, for solving systems of polynomial equations. These methods can handle systems with positive-dimensional and zero-dimensional solution sets, including singular sets. Early work in the area concentrated on solving over the complex number field, where algebraic completeness simplifies matters, but recent work has extended the methods to the real number field. In that realm, most of the effort goes into solving for the boundaries of solution sets as well as their singularities. We will discuss the present status of the area along with some applications, such as mechanism design and the least-squares approximation of measured impedance data by rational polynomials, as is useful in linear control systems.