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Solving Polynomial Systems Equation by Equation

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Algorithms in Algebraic Geometry

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 146))

Abstract

By a numerical continuation method called a diagonal homotopy, one can compute the intersection of two irreducible positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure to intersect general solution sets that are not necessarily irreducible or even equidimensional. Of particular interest is the special case where one of the sets is defined by a single polynomial equation. This leads to an algorithm for finding a numerical representation of the solution set of a system of polynomial equations introducing the equations one by one. Preliminary computational experiments show this approach can exploit the special structure of a polynomial system, which improves the performance of the path following algorithms.

This material is based upon work supported by the National Science Foundation under Grant No. 0105653 and Grant No. 0410047; the Duncan Chair of the University of Notre Dame; the Land Baden-Württemberg (RiP-program at Oberwolfach); and the Institute for Mathematics and Its Applications (IMA), Minneapolis.

This material is based upon work supported by the National Science Foundation under Grant No. 0105739, Grant No. 0134611, and Grant No. 0410036; the Land Baden-Württemberg (RiP-program at Oberwolfach); and the Institute for Mathematics and Its Applications (IMA), Minneapolis

This material is based upon work supported by the National Science Foundation under Grant No. 0410047; General Motors Research and Development; the Land Baden-Württemberg (RiP-program at Oberwolfach); and the Institute for Mathematics and Its Applications (IMA), Minneapolis.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556-4618, USA

    Andrew J. Sommese

  2. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan (M/C 249), Chicago, IL, 60607-7045, USA

    Jan Verschelde

  3. General Motors Research and Development, Mail Code 480-106-359, 30500 Mound Road, Warren, MI, 48090-9055, USA

    Charles W. Wampler

Authors
  1. Andrew J. Sommese
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  2. Jan Verschelde
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  3. Charles W. Wampler
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Editor information

Editors and Affiliations

  1. Departamento de Matemática Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428), Buenos Aires, Argentina

    Alicia Dickenstein

  2. Mathematik und Informatik, Universität des Saarlandes, Campus E2 4, D-66123, Saarbrücken, Germany

    Frank-Olaf Schreyer

  3. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556-4618, USA

    Andrew J. Sommese

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Sommese, A.J., Verschelde, J., Wampler, C.W. (2008). Solving Polynomial Systems Equation by Equation. In: Dickenstein, A., Schreyer, FO., Sommese, A.J. (eds) Algorithms in Algebraic Geometry. The IMA Volumes in Mathematics and its Applications, vol 146. Springer, New York, NY. https://doi.org/10.1007/978-0-387-75155-9_8

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