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A Special Homotopy Continuation Method for a Class of Polynomial Systems

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Computer Algebra in Scientific Computing (CASC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10490))

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Abstract

A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of this method is between the total degree bound and the mixed volume bound and can be easily computed. The new algorithm has been implemented as a program called LPH using C++. Our experiments show its efficiency compared to the polyhedral or other homotopies on such systems. As an application, the algorithm can be used to find witness points on each connected component of a real variety.

The work is partly supported by the projects NSFC Grants 11471307, 11290141, 11271034, 61532019 and CAS Grant QYZDB-SSW-SYS026.

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Acknowledgement

We gratefully acknowledge the very helpful suggestions of Hoon Hong on this paper with emphasis on Sect. 6. We also thank Changbo Chen for his helpful comments. And the authors would like to thank the anonymous reviewers for their constructive comments that greatly helped improving the paper.

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Correspondence to Wenyuan Wu .

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Wang, Y., Wu, W., Xia, B. (2017). A Special Homotopy Continuation Method for a Class of Polynomial Systems. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2017. Lecture Notes in Computer Science(), vol 10490. Springer, Cham. https://doi.org/10.1007/978-3-319-66320-3_26

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  • DOI: https://doi.org/10.1007/978-3-319-66320-3_26

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