Optimal Polyhedral Homotopies on Supercomputers for Algebraic Sets

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Algebraic sets are solutions of polynomial systems. Exploiting the sparse structure of a polynomial system, polyhedral homotopies are optimal for computing all isolated solutions of general systems in the sense that every solution path converges to a solution. The goal of this proposal is to develop new homotopy algorithms that will be generically optimal for computing numerical representations of positive dimensional solution sets. To solve large polynomial systems, the new homotopy algorithms will be made suitable to run on supercomputers.Polynomial systems arise frequently in many problems in science and engineering. The algorithms to solve polynomial systems are implemented in an open source software package PHCpack, available for free on the web.User-friendly interfaces to PHCpack (in computer algebra systems such as Maple (commercial), SAGE (open source), and in scientific software systems such as MATLAB (commercial), Octave (open source)) will continue to benefit many scientists and engineers who solve polynomial systems in their research.The principal investigator teaches his students to transfer mathematical technology through software with an eye towards high performance computing.