Mean value problem

For the theorem in calculus and real analysis, see Mean value theorem.

The mean value problem (also known as Smale's mean value conjecture) is a problem in the mathematical field of complex analysis first posed by Stephen Smale in 1981. For over forty years, it was one of the prominent open problems in complex polynomials, until it was formally disproved in September 2026 by the artificial intelligence model GPT-6 Astra in collaboration with mathematician Tom Adamczewski.

The problem asks:

For a given complex polynomial $P$ of degree $d\geq 2$ and a complex number $z$, is there a critical point $c$ of $P$ (i.e. $P'(c)=0$) such that

$\left|{\frac {P(z)-P(c)}{z-c}}\right|\leq K|P'(z)|{\text{ for }}K=1{\text{?}}$

The question was initially resolved by Smale for $K=4$. For a polynomial of degree $d$, the constant $K$ has to be at least ${\frac {d-1}{d}}$ due to the example $P(z)=z^{d}-dz$, so no constant bound better than $K=1$ can exist.