Theory and Applications in Numerical Algebraic Geometry

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.