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Solving polynomial systems

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Conclusion

The research in this area is in its preliminary stage. Using Theorem A to compute all isolated solutions of a polynomial system requires an amount of computational effort proportional to the product of the degrees —roughly, proportional to the size of the system. Homotopies for solving deficient systems for which the computational effort is instead proportional to the actual number of solutions are being developed. In general, homotopies are constructed to respect the special structure of the deficient system. Theorem C helped to solve problems in dynamical systems [11], the load-flow equations in power systems [9], and various eigenvalue problems [2,8,12]. Many polynomial systems in engineering models are not covered by Theorem C, however; see, for example [14,15,17]. An interesting feature of using the homotopy continuation method to solve polynomial systems is that the curves followed by the scheme are computed independently of one another. Therefore, the algorithm is an excellent candidate for exploiting the advantages of parallel processing—the most important special structure of fifth-generation computers. For example, the homotopy continuation method for calculating eigenvalues and eigenvectors of a matrix is a serious alternative to the currently most popular approach— EISPACK [6]. The preliminary computation results are extremely promising.

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  1. Department of Mathematics, Michigan State University, East Lansing, 48824-1027, MI, USA

    Tien-Yien Li

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Li, TY. Solving polynomial systems. The Mathematical Intelligencer 9, 33–39 (1987). https://doi.org/10.1007/BF03023953

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