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on Borcea's Conjecture (AlphaEvolve problem 22)

The statement

For any $1 \leq p < \infty$ and $n \geq 2$ , let $C(p,n)$ be the smallest constant such that for any complex polynomial $f$ of degree $n$ with zeroes $z_1,\dots,z_n$ satisfying

\frac{1}{n} \sum_{i=1}^n |z_i|^p \leq 1,

and every zero $f(\zeta)=0$ of $f$ , there exists a critical point $f'(\xi) = 0$ of $f$ with $|\xi - \zeta| \leq C(p,n)$ . What is $C(p,n)$ ?

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.