Smale's mean value conjecture

Close the gap between 1 - 1/n and about 4 in Smale's mean value inequality for complex polynomials, or find a counterexample.

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1	# Smale's mean value conjecture
2
3	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.
4
5	## Problem
6
7	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
8
9	$$\left\| \frac{f(z) - f(\xi)}{z - \xi} \right\| \le C(n)\, \|f'(z)\|.$$
10
11	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 $n$.
12
13	## Background
14
15	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.
16
17	## What is known
18
19	- Bounds: $1 - 1/n \le C(n) \le 4$ (Smale, 1981).
20	- 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.
21	- 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).
22	- 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.
23
24	## What would count as progress
25
26	- 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.
27	- An upper bound $C(n) \le c$ with a constant $c < 4$ independent of $n$.
28	- 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](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/MeanValueProblem.lean).
29
30	## References
31
32	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](https://doi.org/10.1090/S0273-0979-1981-14858-8)
33	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](https://doi.org/10.1016/0885-064X(89)90019-8)
34	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](https://doi.org/10.1007/s002080000184)
35	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](https://doi.org/10.1112/blms/bdm063)
36	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](https://arxiv.org/abs/2511.02864)
37
38	## Source
39
40	Imported from [AlphaEvolve Repository of Problems, problem 23](https://google-deepmind.github.io/alphaevolve_repository_of_problems/problems/23.html), 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.