Extremal Polynomials in Smale's Mean Value Conjecture

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Let p be a non-linear complex polynomial in one variable. Smale's mean value conjecture is a precise estimate of the derivative p'(z) in terms of the gradients of chords between z and a stationary point on the graph of p. The problem is to determine the correct constant in the estimate, but despite the apparent simplicity of the problem only a small amount of progress has been made since Stephen Smale first posed it in 1981. In this paper we establish the existence of extremal polynomials for Smale's mean value conjecture, and establish a geometric property of the extremals.
Translated title of the contributionExtremal Polynomials in Smale's Mean Value Conjecture
Original languageEnglish
Pages (from-to)145 - 163
Number of pages19
JournalComputational Methods and Function Theory
Volume6 (1)
Publication statusPublished - Jan 2006

Bibliographical note

Publisher: Heldermann Verlag


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