Crouzeix conjecture

Crouzeix's conjecture asserts that every square complex matrix $A$ and polynomial $p$ satisfy $\|p(A)\|\le 2\sup_{z\in W(A)}|p(z)|$. It is attributed to Michel Crouzeix and has remained open in its general form.

Known results

• Crouzeix proved the general bound $11.08$.
• Crouzeix and Palencia improved it to $1+\sqrt{2}$.
• The conjecture is proved for normal matrices and all $2\times2$ matrices.
• Further proofs cover selected tridiagonal, nilpotent, shift-compression, and spectrally separated classes.

August 2026 claimed proof

Jin Shanmu's GitHub repository provides a standalone LaTeX proof and reports audit results. Public descriptions attribute the result to a $16$-hour autonomous run of GPT-5.6 Sol; an arXiv paper cites Jin's updated preprint as a solution. The claim remains unverified because the proof has no formal peer-review confirmation or independent verification in the retrieved sources.

Current status (as of August 2026): A claimed general proof exists and is attributed to Jin Shanmu with GPT-5.6 Sol, but the conjecture remains mathematically unconfirmed pending independent verification.