Smale's mean value conjecture

For a complex polynomial, compare the slope of a secant from a point $z$ to a critical point with the derivative at $z$. Smale proved in 1981 that some critical point gives a secant slope at most 4 times $|f'(z)|$ and conjectured that the right factor is $1 - 1/n$; after forty-five years the gap is still almost the whole interval.

Problem

For $n \ge 2$, let $C(n)$ be the least constant such that for every complex polynomial $f$ of degree $n$ and every $z \in \mathbb{C}$ with $f'(z) \ne 0$, there is a critical point $\xi$ ($f'(\xi) = 0$) with

$$\left| \frac{f(z) - f(\xi)}{z - \xi} \right| \le C(n)\, |f'(z)|.$$

Establish upper and lower bounds for $C(n)$ that are as strong as possible. Smale's conjecture asserts $C(n) = 1 - 1/n$; the weaker form asks whether $C(n) \le 1$ for all $n$.

Background

Smale (1981) proved the bound with constant 4 in his paper on the fundamental theorem of algebra and complexity theory, and posed the sharp constant as a problem. The polynomial $f(z) = z^n - nz$ at $z = 0$ shows $C(n) \ge 1 - 1/n$: its critical points are the $(n-1)$-th roots of unity, and every secant slope from 0 to them equals $(n-1)/n$ times $|f'(0)|$. The conjecture is known in special situations, for example when all roots of $f - f(z)$ lie on a line through $z$ or have the same distance from $z$ (Tischler, 1989), but the general upper bound has moved only slightly below 4.

What is known

• Bounds: $1 - 1/n \le C(n) \le 4$ (Smale, 1981).
• Improved upper bounds: degree-dependent improvements by Beardon, Minda and Ng (2002), Fujikawa and Sugawa (2006), and Conte, Fujikawa and Lakic (2007); for $n \ge 8$, $C(n) < 4 - 2.263/\sqrt{n}$ (Crane, 2007). Georgiev et al. describe these as slight improvements; Crane's bound tends to 4 as $n$ grows.
• Special cases: the conjectured bound $1 - 1/n$ holds at points $z$ for which the roots of $f - f(z)$ are collinear with $z$ or equidistant from it (Tischler, 1989).
• Numerics: Georgiev, Gómez-Serrano, Tao and Wagner (2025) searched over root configurations and recovered $z^n - nz$ and other configurations with the same value, but found no counterexample.

What would count as progress

• A counterexample: a polynomial $f$ and a point $z$ for which every critical point $\xi$ gives a ratio above $1 - 1/n$. This is checkable in milliseconds with exact or interval arithmetic: compute the critical points and evaluate the ratio for each.
• An upper bound $C(n) \le c$ with a constant $c < 4$ independent of $n$.
• A proof of the conjecture for a new class of polynomials or for small degrees. A Lean statement of the weak form is in formal-conjectures.

References

1. S. Smale, "The fundamental theorem of algebra and complexity theory", Bulletin of the AMS 4 (1981), 1–36. doi:10.1090/S0273-0979-1981-14858-8
2. D. Tischler, "Critical points and values of complex polynomials", Journal of Complexity 5 (1989), 438–456. doi:10.1016/0885-064X(89)90019-8
3. A. F. Beardon, D. Minda and T. W. Ng, "Smale's mean value conjecture and the hyperbolic metric", Mathematische Annalen 322 (2002), 623–632. doi:10.1007/s002080000184
4. E. Crane, "A bound for Smale's mean value conjecture for complex polynomials", Bulletin of the London Mathematical Society 39 (2007), 781–791. doi:10.1112/blms/bdm063
5. B. Georgiev, J. Gómez-Serrano, T. Tao and A. Z. Wagner, "Mathematical exploration and discovery at scale" (2025), Problem 6.23. arXiv:2511.02864

Source

Imported from AlphaEvolve Repository of Problems, problem 23, curated by Google DeepMind. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.