Solving Polynomial Systems Equation by Equation

  • Andrew J. Sommese
  • Jan Verschelde
  • Charles W. Wampler
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 146)


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.

Key words

Algebraic set component of solutions diagonal homotopy embedding equation-by-equation solver generic point homotopy continuation irreducible component numerical irreducible decomposition numerical algebraic geometry path following polynomial system witness point witness set 

AMS(MOS) subject classifications

Primary 65H10 Secondary 13P05 14Q99 68W30 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Andrew J. Sommese
    • 1
  • Jan Verschelde
    • 2
  • Charles W. Wampler
    • 3
  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA
  2. 2.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  3. 3.General Motors Research and DevelopmentWarrenUSA

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