Extremal Problems for Polynomials in the Complex Plane

  • Borislav Bojanov
Part of the Springer Optimization and Its Applications book series (SOIA, volume 42)


This is a survey on some particular polynomial problems that are related to complex analogs of Rolle’s theorem or to the Bernstein majorization theorem that implies the well-known estimate for the derivative of a complex polynomial on the disk. The main topic, however, is Sendov’s conjecture about the critical points of algebraic polynomials. Despite the numerous attempts to verify the conjecture, it is not settled yet and remains as one of the most challenging problems in the analytic theory of polynomials. We also discuss the mean value conjecture of Smale and point out to certain relation between these two famous open problems. Finally, we formulate a conjecture that seems to be a natural complex analog of Rolle’s theorem and contains as a particular case Smale’s conjecture.


Complex Plane Unit Disk Extremal Problem Critical Radius Algebraic Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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The author is grateful to his colleagues Lozko Milev and Nikola Naidenov for their help in performing computer calculations confirming Conjecture 1 for polynomials of small degree.

This work was supported by the Sofia University Research Grant # 135/2008 and by Swiss-NSF Scopes Project IB7320-111079.


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Borislav Bojanov
    • 1
  1. 1.Department of MathematicsUniversity of SofiaSofiaBULGARIA

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