Numerical algebraic geometry involves the use of numerical methods to extract data from ideals about the corresponding varieties or schemes. This area has grown rapidly over the last 20 years and has found applications in many areas of science and engineering. This grant is funding four projects in the area of numerical algebraic geometry. First, the PI and his students will investigate several forms of preconditioning for homotopy continuation, including algorithms for finding (near-)optimal multihomogeneous and linear product start systems, as well as the use of dual bases to reduce the number of paths tracked to "bad" endpoints. Second, the PI and several collaborators will work on three application areas: exceptional mechanisms (via fiber products), software in Macaulay2 for algebraic geometry-related applications, and software for repeated parameter homotopies. Third, the PI will work on analyzing the complexity of numerical algebraic geometry algorithms. Finally, the PI and several collaborators will continue to work on methods to extract information about real algebraic sets both from standard continuation methods and from Khovanskii-Rolle continuation.
Polynomial systems of equations arise in many places throughout mathematics, science, and engineering. An entire mathematical field - algebraic geometry - grew out of the need to find solutions to these sorts of equations. Until the 1960s, though, there was no known general technique for solving such systems of equations. The methods developed in that era require too much memory to be effective except for relatively small problems. More recently developed methods - the numerical methods of Sommese, Verschelde, Wampler, Li, and others, now collectively known as numerical algebraic geometry - allow for the solution of much larger polynomial systems, opening the application of algebraic geometry methods to a wider class of problems. However, there is still much to understand about these numerical methods. The goals of this project include addressing four open problems in this direction. This work includes the development of techniques to streamline some of these computations, the implementation of valuable algorithms in popular and useful software packages, a careful analysis of the computational costs associated with the computational methods in this field, and the continued effort to extract useful real-world data from the data provided as output from these methods.