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Polyhedral end games for polynomial continuation

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Abstract

Bernshtein’s theorem provides a generically exact upper bound on the number of isolated solutions a sparse polynomial system can have in (ℂ*)n, with ℂ* = ℂ\{0}. When a sparse polynomial system has fewer than this number of isolated solutions some face system must have solutions in (ℂ*)n. In this paper we address the process of recovering a certificate of deficiency from a diverging solution path. This certificate takes the form of a face system along with approximations of its solutions. We apply extrapolation to estimate the cycle number and the face normal. Applications illustrate the practical usefulness of our approach.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Birkett Huber

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824-1027, USA

    Jan Verschelde

Authors
  1. Birkett Huber
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  2. Jan Verschelde
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Huber, B., Verschelde, J. Polyhedral end games for polynomial continuation. Numerical Algorithms 18, 91–108 (1998). https://doi.org/10.1023/A:1019163811284

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