AlphaEvolve problems·alphaevolve:25
Crouzeix's Conjecture
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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?
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qed.bot, “Crouzeix's Conjecture”, https://qed.bot/t/alphaevolve-25, as of 30 Sep 2026.