Dual Smale's mean value conjecture for odd polynomials

Quanyu Tang

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arXiv · 2510.16875

Dual Smale's mean value conjecture for odd polynomials

Quanyu Tang

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Abstract

We prove Dual Smale's mean value conjecture for all odd polynomials with nonzero linear term. Precisely, if $P$ is an odd polynomial of degree $d\ge3$ with $P(0)=0$ and $P'(0)=1$, then there exists a critical point $\zeta$ of $P$ such that $$ \left|\frac{P(\zeta)}{\zeta}\right| \ge \frac1d. $$This result can be regarded as a dual counterpart of T. W. Ng's theorem on Smale's mean value conjecture for odd polynomials with nonzero linear term [J. Aust. Math. Soc. 75 (2003), 409--411].

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Quanyu Tang. 2025-10-19. Dual Smale's mean value conjecture for odd polynomials. https://arxiv.org/abs/2510.16875

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