Crouzeix's Conjecture

Crouzeix conjectured in 2004 that for every square complex matrix $A$ and every polynomial $p$, $\lVert p(A)\rVert \leq 2 \max_{z \in W(A)} |p(z)|$, where $W(A)$ is the numerical range of $A$ - that is, the numerical range is a 2-spectral set. Crouzeix proved a constant of 11.08 in 2007 and Crouzeix and Palencia lowered it to $1+\sqrt{2}$ in 2017; the conjectured constant 2 is attained by $2\times 2$ matrices. Jin proves the sharp bound by a function-theoretic route whose key theorem reduces the problem, via a sampling strategy, to a positivity condition; Lorist and Schwenninger independently prove it days later by combining double-layer potential machinery with a perturbation lemma for 2-dilations.

Two independent proofs within eight days, both with AI in the loop. Jin's (posted 27 July, preprints.org, submitted to Annals) is the first: its decisive theorem came out of an autonomous GPT-5.6 Sol run, and it is the proof Townsend, Greenbaum and Crouzeix have checked. Lorist and Schwenninger's five-page argument (arXiv, 4 August) is a genuinely different route - double-layer potentials plus a perturbation lemma for 2-dilations - produced with ChatGPT 5.6 Pro exploring proof strategies. The entry's headline axes record Jin's proof; the earlier version of this entry recorded Lorist-Schwenninger's as primary while Jin's AI provenance was still unknown.