Mean value problem

In mathematics, the mean value problem was posed by Stephen Smale in 1981.^[1] This problem is still open in full generality. The problem asks:

Given a complex polynomial ƒ and a complex number z, is there a critical point c of ƒ (i.e. ƒ ′(c) = 0) such that

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

It was proved for K = 4 (see the article cited above). For a polynomial of degree d the constant K has to be at least (d − 1)/d from the example f(z) = z^d − dz, therefore no bound better than K = 1 can exist. Tischler has some partial results.

References

1. "The Fundamental Theorem of Algebra and Complexity Theory", Bulletin (New Series) of the American Mathematical Society, volume 4, number 1, 1981.

External links

• The Fundamental Theorem of Algebra and Complexity Theory, by Stephen Smale