Counterexamples to Tischler's Strong Form of Smale's Mean Value Conjecture |
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Authors: | Tyson Jeremy T. |
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Affiliation: | Department of Mathematics, University of Illinois 1409 W. Green St., Urbana, IL 61801 USA; tyson{at}math.uiuc.edu |
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Abstract: | Smale's mean value conjecture asserts that for every polynomial P of degree d satisfying P(0)=0,where K = (d1)/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)=zd 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. |
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