Submit progress

on Smale's Problem (AlphaEvolve problem 23)

The statement

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

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

Establish upper and lower bounds for $C(n)$ that are as strong as possible.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.