Mesh-Expanding Homotopies and Numerical Irreducible Decomposition Over a Number Field

Algorithms from the field of numerical algebraic geometry provide robust means to compute all isolated solutions of arbitrary systems of polynomials and to give a thorough numerical description of positive-dimensional solution components. However, the number of isolated complex solutions tends to grow exponentially with respect to the number $N$ of equations and variables, limiting the size of tractable systems. This is a particular problem for systems arising from discretizations of ordinary differential equations because $N$ is directly related to mesh size, and finer meshes are generally more desirable. On the other hand, the number of interesting solutions is often far smaller than the total count, and much effort is wasted on complex solutions with no physical interpretation.In the first part of this thesis, a general method is given for constructing a homotopy that directly relates solutions on grids of different resolutions. This enables iterative generation of solutions on increasingly fine meshes, and makes it possible to filter out some non-physical solutions to substantially reduce wasted computation time. In the second part, a new algorithm is given to compute the numerical irreducible decomposition of a general polynomial system over a finite algebraic extension of $Q$ by projecting the solution set onto a line and deducing the exact defining polynomial of the projection.