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Solving Polynomial Systems Using Numeric Gröbner Bases

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Mathematical Software – ICMS 2018 (ICMS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10931))

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Abstract

Systems of polynomial or algebraic equations with finitely many solutions arise in many areas of applied mathematics. I will discuss the design and implementation of a hybrid symbolic-numeric method based on the endomorphism matrix approach pioneered by Stetter and others. It makes use of numeric Gröbner bases and arbitrary-precision eigensystem computations. I will describe how to assess accuracy, find and remove parasite solutions in the case of fractional degrees in the system, handle multiplicity, as well as some of the other finer points not usually covered in the literature. This work is one of the methods used in the Wolfram Language NSolve function.

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Author information

Authors and Affiliations

  1. Wolfram Research, Champaign, IL, USA

    Daniel Lichtblau

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  1. Daniel Lichtblau
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Correspondence to Daniel Lichtblau .

Editor information

Editors and Affiliations

  1. University of Bath, Bath, United Kingdom

    James H. Davenport

  2. Johannes Kepler University, Linz, Austria

    Manuel Kauers

  3. University of Waterloo, Waterloo, Ontario, Canada

    George Labahn

  4. Czech Technical University in Prague, Prague 6, Czech Republic

    Josef Urban

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Lichtblau, D. (2018). Solving Polynomial Systems Using Numeric Gröbner Bases. In: Davenport, J., Kauers, M., Labahn, G., Urban, J. (eds) Mathematical Software – ICMS 2018. ICMS 2018. Lecture Notes in Computer Science(), vol 10931. Springer, Cham. https://doi.org/10.1007/978-3-319-96418-8_40

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  • DOI: https://doi.org/10.1007/978-3-319-96418-8_40

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-96417-1

  • Online ISBN: 978-3-319-96418-8

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