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Crouzeix's Conjecture (AlphaEvolve problem 25)

Crouzeix's Conjecture (AlphaEvolve problem 25). Current research status is unassessed; the repository records AlphaEvolve’s result on this question.

MSC 47A12 · Numerical range, numerical radiusMSC 15A60 · Norms of matrices, numerical range, applications of functional analysis to matrix theory

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository at commit 8f44745 (2026-07-11); the repository records AlphaEvolve results, not current research statusSource & evidence ↓
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Problem statement

Let CC be the smallest constant for which one has the bound

∥p(A)∥op≤Csup⁡z∈W(A)∣p(z)∣\| p(A) \|_{op} \leq C \sup_{z \in W(A)} |p(z)|

for all n×nn \times n square matrices AA and all polynomials pp with complex coefficients, where ∥∥op\| \|_{op} is the operator norm and

W(A)≔{⟨Ax,x⟩:∥x∥≤1}W(A) \coloneqq \{ \langle Ax, x \rangle: \|x\| \leq 1 \}

is the numerical range of AA. What is CC? What polynomials pp attain the bound with equality?

Lean 4

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Background

The repository status chart does not categorize this problem. See Section 6.10, in "Mathematical Exploration and Discovery at Scale " paper: arXiv:2511.02864. Experiment notebook.

Progress & source assessment

The repository page records no bounds for this problem.

References
  1. [1]Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Repository of problems accompanying “Mathematical exploration and discovery at scale” (arXiv:2511.02864), github.com/google-deepmind/alphaevolve_repository_of_problems (commit 8f44745, 11 July 2026). — Crouzeix's Conjecture (AlphaEvolve problem 25) linkApache-2.0 (repository license); problem pages reproduced with attribution.

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