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on Crouzeix's Conjecture (AlphaEvolve problem 25)

The statement

Let $C$ be the smallest constant for which one has the bound

\| p(A) \|_{op} \leq C \sup_{z \in W(A)} |p(z)|

for all $n \times n$ square matrices $A$ and all polynomials $p$ with complex coefficients, where $\| \|_{op}$ is the operator norm and

W(A) \coloneqq \{ \langle Ax, x \rangle: \|x\| \leq 1 \}

is the numerical range of $A$ . What is $C$ ? What polynomials $p$ attain the bound with equality?

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.