Counterexamples to Tischler's Strong Form of Smale's Mean Value Conjecture
Department of Mathematics, University of Illinois 1409 W. Green St., Urbana, IL 61801 USA; tyson{at}math.uiuc.edu
Received 14 February 2003. Revision received 7 November 2003.
Smale's mean value conjecture asserts that 
for every polynomial P of degree d satisfying P(0)=0, where K = (d–1)/d and the minimum is taken over all critical points 
of P. A stronger conjecture due to Tischler asserts that
 |
with 
. Tischler's conjecture is known to be true: (i) for local perturbations of the extremum P0(z)=zd – dz, and (ii) for all polynomials of degree d 
4. In this paper, Tischler's conjecture is verified for all local perturbations of the extremum P1(z)=(z – 1)d – (–1)d, but counterexamples to the conjecture are given in each degree d 
5. In addition, estimates for certain weighted L1- and L2-averages of the quantities 
are established, which lead to the best currently known value for K1 in the case d=5. 2000 Mathematics Subject Classification 30C15.
Research supported by the U.S. National Science Foundation under Grant DMS-0228807.