Crouzeix's Conjecture
Crouzeix conjectured in 2004 that for every square complex matrix and every polynomial , , where is the numerical range of . Crouzeix proved a constant of 11.08 in 2007 and Crouzeix and Palencia lowered it to in 2017; the conjectured constant 2 is attained by matrices. Lorist and Schwenninger prove the sharp bound by combining the double-layer potential machinery with a perturbation lemma for 2-dilations.
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-assisted
- Method
- Argument
- Field
- Matrix analysis
- Posed by
- Michel Crouzeix
- Year posed
- 2004
- Years open
- 22y
- Solved
- 2026-08-04
- Model
- ChatGPT 5.6 Pro
- Vendor
- OpenAI
- Collaborators
- Emiel Lorist, Felix L. Schwenninger
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- 2 languages
What was actually shown
Concurrent independent proof: the authors note that during preparation of this note, a proof of the conjecture appeared independently in a July 2026 preprint of S. Jin ("The numerical range is a 2-spectral set", preprints.org), by a different, function-theoretic route. That work carries no AI disclosure known to us, so the first proof of Crouzeix's conjecture may not be an AI-assisted one; this entry records the AI-assisted proof, not a claim of priority.
What the AI did
The paper's disclosure in full: "ChatGPT 5.6 Pro by OpenAI was used to explore proof strategies for this note. The note was entirely written by the authors, who take full responsibility for the content." No individual lemma is attributed to the model, so the lower tier applies.
Verification
A preprint days old, resolving a conjecture open since 2004. No independent expert review, no formalization, and a competing independent claim is outstanding (see the result note), so this is recorded as a candidate rather than a settled result.
Sources
- Independent workJin's concurrent independent proof (preprints.org, July 2026)
Submitted by NimbleRaven553 on