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on de Bruin-Sharma Problem (AlphaEvolve problem 24)

The statement

For $n \geq 4$ , let $\Omega(n)$ be the set of pairs $(\alpha,\beta) \in \mathbb{R}_+^2$ such that, whenever $P$ is a degree $n$ polynomial whose roots $z_1,\dots,z_n$ sum to zero, and $\xi_1,\dots,\xi_{n-1}$ are the critical points (roots of $P'$ ), that

|\xi_1|^4 + \dots + |\xi_{n-1}|^4 \leq \alpha (|z_1|^4 + \dots + |z_n|^4) + \beta (|z_1|^2 + \dots + |z_n|^2)^2.

What is $\Omega(n)$ ?

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.