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Symbolic-Numeric Computation pp 223–243Cite as

Pythagore’s Dilemma, Symbolic-Numeric Computation, and the Border Basis Method

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Part of the book series: Trends in Mathematics ((TM))

Abstract

In this tutorial paper, we first discuss the motivation of doing symbolic-numeric computation, with the aim of developing efficient and certified polynomial solvers. We give a quick overview of fundamental algebraic properties, used to recover the roots of a polynomial system, when we know the multiplicative structure of its quotient algebra. Then, we describe the border basis method, justifying and illustrating the approach on several simple examples. In particular, we show its usefulness in the context of solving polynomial systems, with approximate coefficients. The main results are recalled and we prove a new result on the syzygies, naturally associated with commutation properties. Finally, we describe an algorithm and its implementation for computing such border bases.

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Author information

Authors and Affiliations

  1. GALAAD, INRIA, BP 93, 06902, Sophia Antipolis, France

    Bernard Mourrain

Authors
  1. Bernard Mourrain
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Editor information

Editors and Affiliations

  1. School of Science, Beihang University, 37 Xueyuan Road, Beijing, 100083, China

    Dongming Wang

  2. Laboratoire d’Informatique de Paris 6, Université Pierre et Marie Curie - CNRS, 8, rue due Capitaine Scott, 75015, Paris, France

    Dongming Wang

  3. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and System Sciences, Beijing, 100080, China

    Lihong Zhi

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© 2007 Birkhäuser Verlag Basel/Switzerland

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Mourrain, B. (2007). Pythagore’s Dilemma, Symbolic-Numeric Computation, and the Border Basis Method. In: Wang, D., Zhi, L. (eds) Symbolic-Numeric Computation. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7984-1_14

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