COUNTEREXAMPLES TO TISCHLER'S STRONG FORM OF SMALE'S MEAN VALUE CONJECTURE

Abstract

Smale's mean value conjecture asserts that $\min_\theta |P(\theta)/\theta| \le K|P'(0)|$ 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 $\theta$ of $P$. A stronger conjecture due to Tischler asserts that \[ \min_\theta\left|\frac12-\frac{P(\theta)}{\theta\cdot P'(0)}\right| \le K_1 \] with $K_1=\frac12-1/d$. Tischler's conjecture is known to be true: (i) for local perturbations of the extremum $P_0(z)=z^d-dz$, and (ii) for all polynomials of degree $d\le 4$. In this paper, Tischler's conjecture is verified for all local perturbations of the extremum $P_1(z)=(z-1)^d-(-1)^d$, but counterexamples to the conjecture are given in each degree $d\ge 5$. In addition, estimates for certain weighted $L^1$- and $L^2$-averages of the quantities $\frac12-{P(\theta)}/{\theta\cdot P'(0)}$ are established, which lead to the best currently known value for $K_1$ in the case $d=5$.