Theory and Applications in Numerical Algebraic Geometry

by Daniel James Bates Homotopy continuation techniques may be used to approximate all isolated solutions of a polynomial system. More recent methods which form the crux of the young field known as numerical algebraic geometry may be used to produce a description of the complete solution set of a polynomial system, including the positive-dimensional solution components. There are four main topics in the present thesis: three novel numerical methods and one new software package. The first algorithm is a way to increase precision as needed during homotopy continuation path tracking in order to decrease the computational cost of using high precision. The second technique is a new way to compute the scheme structure (including the multiplicity and a bound on the Castelnuovo-Mumford regularity) of an ideal supported at a single point. The third method is a new way to approximate all solutions of a certain class of two-point boundary value problems based on homotopy continuation. Finally, the software package, Bertini, may be used for many calculations in numerical algebraic geometry, including the three new algorithms described above. For Sarah, my precious bride.