An introduction to Smale's mean value

conjecture for polynomials


Dr. T.W. Ng

The University of Hong Kong



In 1981, Stephen Smale proved the following interesting result.

Theorem. Let P be a non-linear polynomial of degree d with critical points Image22 If z is not a critical point of P, then there exists some critical point zi 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 number (d - 1)/d would, if true, be the best possible bound here as it is attained (for any nonzero l ) when P(z) = zd - lz and z = 0 in (1.1). The conjecture has only been verified for d = 2, 3, 4 and the general case remains open. In this talk we shall give a general introduction to this conjecture.


October 27, 2000 (Friday) 


4:00 - 5:00pm


Room 517, Meng Wah Complex

All are welcome