The complete Crouzeix inequality

OpenAI's theorem (2026) that ||P[A]|| <= 2 max over the numerical range W(A) of ||P(z)|| for every complex square matrix A and every polynomial P with matrix coefficients of any size, with the sharp constant 2: the complete form of Crouzeix's conjecture. Convex domains with analytic polar boundaries approximate W(A) from outside; the Riemann map of the inverted exterior extends across the boundary; the Faber polynomials of the domain give a nonnegative kernel and a two-sided estimate of boundary norms by coefficients; the resolvent has positive Hermitian part on the circle, and testing it with the singular vectors of F[A] and the products FG and GF gives the constant 2.

Five lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. The numerical range and the Crouzeix inequality
2. Smooth convex neighbourhoods
3. An exterior conformal coordinate
4. Faber polynomials and a positive kernel
5. The bound on a smooth convex domain

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Linear algebra over the complex numbers: tensor products of matrices, positive semidefinite matrices, the Hilbert–Schmidt norm and singular vectors.
• Complex analysis as in the core course Complex Analysis, cited by number at each use: Cauchy's formula and estimates, Rouché's theorem, the Riemann mapping theorem, harmonic conjugates and the reflection principle for harmonic functions.