Mathematical Sciences: Computation Using Homotopies and Ordered Fields

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This research explores two closely related methods for solving equations, the use of homotopies and the use of infinitesimals, in the context of ordered fields and piecewise linearity. The homotopy approach involves using a single parameter to deform a system of equations into an auxiliary and more easily solved system, then tracing the solution back to the solution of the original system. Applications include problems in economics and engineering. Specifically, an economic model with incomplete markets will be analyzed. The resulting system of equations has a Grassmanian manifold as the domain -- a feature which interjects new obstacles and opportunities. The infinitesimal approach involves solutions of a parametrized family of systems. By replacing the parameter with an infinitesimal variable and solving the system relative to a larger field containing the infinitesimal, a family of solutions are obtained in an efficient way. Applications include linear inequality systems, linear programming problems, bounded variable integer programs, linear complementary problems, and quadratic programs.