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on Sendov's Conjecture (AlphaEvolve problem 20)

The statement

For each $n \geq 2$ , let $C(n)$ be the smallest constant such that for any complex polynomial $f$ of degree $n \geq 2$ with zeros $z_1, \dots, z_n$ in the unit disk and critical points $w_1, \dots, w_{n-1}$ ,

\max_{1 \leq k \leq n} \min_{1 \leq j \leq n-1} |z_k - w_j| \leq C(n).

Sendov conjectured that $C(n)=1$ .

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.