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Certifying Simple Zeros of Over-Determined 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

We construct a real square system related to a given over-determined real system. We prove that the simple real zeros of the over-determined system are the simple real zeros of the related square system and the real zeros of the two systems are one-to-one correspondence with the constraint that the value of the sum of squares of the polynomials in the over-determined system at the real zeros is identically zero. After certifying the simple real zeros of the related square system with the interval methods, we assert that the certified zero is a local minimum of the sum of squares of the input polynomials. If the value of the sum of the squares of the input polynomials at the certified zero is equal to zero, then it is a zero of the input system. Notice that a complex system with complex zeros can be transformed into a real system with real zeros.

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Acknowledgement

The work is partially supported by NSFC Grants 11471327.

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Correspondence to Jin-San Cheng or Xiaojie Dou .

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Cheng, JS., Dou, X. (2017). Certifying Simple Zeros of Over-Determined 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_6

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  • DOI: https://doi.org/10.1007/978-3-319-66320-3_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66319-7

  • Online ISBN: 978-3-319-66320-3

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