| The topics in this dissertation, while independent, are unified under the field of numerical algebraic geometry. With ties to some of the oldest areas in mathematics, numerical alge- braic geometry is relatively young as a field of study in its own right. The field is concerned with the numerical approximation of the solution sets of systems of polynomial equations and the manipulation of these sets. Given a polynomial system f : CN → C n, the methods of numerical algebraic geometry produce numerical approximations of the isolated solutions of f(z) = 0, as well as points on any positive-dimensional components of the solution set, V(f). In a short time, the work done in numerical algebraic geometry has significantly pushed the boundary of what is computable. This dissertation aims to further this work by contributing new algorithms to the field and using cutting edge techniques of the field to expand the scope of problems that can be addressed using numerical methods. We begin with an introduction to numerical algebraic geometry and subsequent chapters address independent topics: perturbed homotopies, exceptional sets and fiber products, and a numerical approach to finding unit distance embeddings of finite simple graphs. (Abstract shortened by UMI.). |
