Let P∈C[mathbfz] be a polynomial of degree n+1≥ 2 such that P(0)=0 and P'(0)=1, and let b_1,…,b_n be its critical points. Smale's mean value conjecture. min|fracP(b_i)b_i|:i=1,…,n≤ fracnn+1. This is equivalent to Smale's question of replacing the constant 4 in his mean value theorem by n/(n+1). It is verified for n=1,2,3, and the upper bound is attained by P(z)=z^n+1+z; the general case remains open.

Smale's mean value conjecture

Smale proposed the conjecture in 1981: for a normalized complex polynomial, at least one critical point should satisfy the sharp degree-dependent bound. The extremal example is known, but the general assertion is unresolved.

Known results

The conjecture is proved in degrees $2$ , $3$ , and $4$ , corresponding to $n=1,2,3$ in the problem's indexing.
Computer calculations verified it through degree $10$ .
Crane obtained the upper bound $M_n<4-\frac{2.263}{\sqrt n}$ for $n\ge 8$ .

August 27, 2026 asymptotic progress

On August 27, 2026, Smale's Mean Value Conjecture and its Dual Conjecture for Complex Polynomials reported asymptotic validity of the conjectured bound within an explicit family as the degree tends to infinity. This is genuine partial progress, but it does not settle the conjecture for all complex polynomials and remains unverified in this report.

Current status (as of August 2026): The conjecture is settled for degrees $2$ , $3$ , and $4$ and computationally checked through degree $10$ ; an asymptotic-family result has been reported, but the unrestricted problem remains open.

Smale's mean value conjecture

Primary source

Progress summary

Solutions 0