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Certifying Reality of Projections

  • Jonathan D. HauensteinEmail author
  • Avinash Kulkarni
  • Emre C. Sertöz
  • Samantha N. Sherman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10931)

Abstract

Computational tools in numerical algebraic geometry can be used to numerically approximate solutions to a system of polynomial equations. If the system is well-constrained (i.e., square), Newton’s method is locally quadratically convergent near each nonsingular solution. In such cases, Smale’s alpha theory can be used to certify that a given point is in the quadratic convergence basin of some solution. This was extended to certifiably determine the reality of the corresponding solution when the polynomial system is real. Using the theory of Newton-invariant sets, we certifiably decide the reality of projections of solutions. We apply this method to certifiably count the number of real and totally real tritangent planes for instances of curves of genus 4.

Keywords

Certification Alpha theory Newton’s method Real solutions Numerical algebraic geometry 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Jonathan D. Hauenstein
    • 1
    Email author
  • Avinash Kulkarni
    • 2
  • Emre C. Sertöz
    • 3
  • Samantha N. Sherman
    • 1
  1. 1.Department of Applied and Computational Mathematics and StatisticsUniversity of Notre DameNotre DameUSA
  2. 2.Department of MathematicsSimon Fraser UniversityBurnabyCanada
  3. 3.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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