Collaborative Research: Numerical Algorithms and Software for Solving Polynomial Systems with Parameters

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In this collaborative project the investigators Sommese,Wampler, and Verschelde construct and implement new algorithms tonumerically describe algebraic sets defined by systems ofparameterized polynomial equations. The scope of work goesbeyond conventional problems, where one seeks the solution setwhen the parameters are given, to include the determination ofparameter sets that change the nature of solution components in aspecified manner. Of special interest are solution sets whosedimension is exceptionally large. Problems such as these, e.g.,the discovery and classification of overconstrained mechanisms,can be reduced to computing the irreducible decomposition ofcertain exceptional loci of polynomial systems with parameters. This is then reduced to finding the numerical irreducibledecomposition of a finite number of associated polynomialsystems. As this leads to large systems, a newequation-by-equation approach to solving polynomial systems isdeveloped. To tackle nontrivial problems, the algorithms isimplemented on parallel computers. Solving a polynomial equation -- finding the roots of thepolynomial -- is an old problem that occurs over and over in morecomplicated forms throughout science and engineering. Inapplications involving mechanical design, finding the solution ofa system of polynomial equations is only one step in a largerscheme: to arrive at better designs by describing how the set ofsolutions of a polynomial system depends on parameters of thesystem. This problem arises in design of industrial robots, forexample. The investigators develop new methods for describingthe solution sets of polynomial systems with parameters. Themethods have the potential to become a standard tool in thedesign of robots and mechanisms. Furthermore, the methodspromise to have a wider impact on the research fields ofnumerical analysis and computer algebra, especially in effortsseeking to provide the scientific community with software tosolve mathematical problems. An important result of this work ispublicly available software based on these new methods, withinterfaces facilitating use by the wider community. By solvingsome difficult polynomial systems that arise in science andengineering, the team stimulates interest in these advancedmethods and provides illustration of their usage for thenonspecialist. The project includes collaboration with andtraining of students.