Extremal Problems for Polynomials in the Complex Plane

Borislav Bojanov
Additional contact information
Borislav Bojanov: University of Sofia

A chapter in Approximation and Computation, 2010, pp 61-85 from Springer

Abstract: Abstract 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.

Keywords: Complex Plane; Unit Disk; Extremal Problem; Critical Radius; Algebraic Polynomial (search for similar items in EconPapers)
Date: 2010
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-1-4419-6594-3_5

Ordering information: This item can be ordered from
http://www.springer.com/9781441965943

DOI: 10.1007/978-1-4419-6594-3_5

Access Statistics for this chapter

More chapters in Springer Optimization and Its Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().