Counterexamples to Tischler's Strong Form of Smale's Mean Value Conjecture

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 criticalpoints of P. A stronger conjecture due to Tischler assertsthat with . Tischler's conjecture is known to be true: (i) for local perturbations of the extremumP₀(z)=z^d – dz, and (ii) for all polynomials of degreed 4. In this paper, Tischler's conjecture is verified for alllocal perturbations of the extremum P₁(z)=(z – 1)^d –(–1)^d, but counterexamples to the conjecture are givenin each degree d 5. In addition, estimates for certain weightedL¹- and L²-averages of the quantities are established, which lead to the best currentlyknown value for K₁ in the case d=5. 2000 Mathematics SubjectClassification 30C15.