Geometry and Analysis Seminar


Smale's mean value conjecture and the hyperbolic metric

Dr. T.W. Ng

The University of Hong Kong


The following interesting result was proved by Steve Smale in 1981.

Theorem. Let P be a polynomial of degree d ?2 and a be any given complex number, then there exists some critical point b of P such that


Smale then asked whether one can replace the factor 4 in the upper bound in (1.1) by 1, or even possibly by (d - 1)/d. The conjecture has been verified for d = 2, 3, 4, and also in some other special circumstances but the general case remains open.

In this talk we shall show by using the method of hyperbolic metric that the constant 4 can be replaced by 4(d-2)/(d-1), where d is the degree of the polynomial. This first improvement of Smale's theorem is a joint work with Alan F. Beardon and David Minda.


November 1, 2000 (Wednesday)


4:00 - 5:00pm


Room 517, Meng Wah Complex




All are welcome