The first part of this thesis is concerned with the problem of solving polynomial equation systems, a problem of great importance in scientific and engineering computing. We propose a new method for solving systems of multivariate nonlinear polynomial equations efficiently and robustly. This method is a symbolic-numerical method that consists in rewriting the system as a single eigenvalue problem. It can be viewed as an extension of the pseudo-inverse approach to solve linear systems of equations. Like the pseudo-inverse approach, our proposed method can handle well-determined systems as well as over-determined systems with noisy coefficients. This makes it particularly applicable, as many polynomial systems encountered in practice are over-determined and noisy. We begin by presenting a solution method for systems of univariate polynomials before generalizing the method to the case of systems of multivariate polynomials.

The second part of the this thesis is concerned with the problem of structure from motion. Structure from motion (SFM) is the problem of recovering the geometry of a scene from a stream of images taken from unknown viewpoints. One popular approach to estimating the geometry of a scene is to track scene features on several images and reconstruct their position in 3D. During this process, the unknown camera pose must also be recovered. Unfortunately recovering the pose is an ill-conditioned problem which, in turn, makes the SFM problem difficult to solve accurately. We propose an alternative formulation of the SFM problem, obtained by algebraic variable elimination, in which the camera pose parameters do not appear. As a result, the problem is better conditioned in addition to involving much fewer variables. Variable elimination is done in three steps. First, we take the standard SFM equations in projective coordinates and eliminate the camera orientations from the equations. We then further eliminate the camera center positions. Finally, we also eliminate all 3D point positions coordinates, except for their depths with respect to the camera center (i.e., the fourth projective coordinates), thus obtaining a set of simple polynomial equations of degree two and three. We show that, when there are merely a few points and pictures, these "depth-only equations" can be solved in a global fashion using homotopy methods. We also show that, in general, these same equations can be used to formulate a cost function to refine SFM solutions in a way that is more accurate than by minimizing the total reprojection error, as done when using the bundle adjustment method.