{
  "id": "the-complete-crouzeix-inequality",
  "title": "The complete Crouzeix inequality",
  "description": "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.",
  "units": [
    {
      "id": "the-numerical-range-and-the-crouzeix-inequality",
      "title": "The numerical range and the Crouzeix inequality",
      "reader": "the-numerical-range-and-the-crouzeix-inequality.html",
      "source": "src/the-numerical-range-and-the-crouzeix-inequality.md",
      "source_sha256": "3eb0f9f21ca5431743b75668cbefca76cbb2293daa14c529deefa324b5c4695a",
      "status": "author_self_checked"
    },
    {
      "id": "smooth-convex-neighbourhoods",
      "title": "Smooth convex neighbourhoods",
      "reader": "smooth-convex-neighbourhoods.html",
      "source": "src/smooth-convex-neighbourhoods.md",
      "source_sha256": "6b86f77d4fe8d1843d8019362af93754bf58e41077f4de1ea05de341e70ab7e4",
      "status": "author_self_checked"
    },
    {
      "id": "an-exterior-conformal-coordinate",
      "title": "An exterior conformal coordinate",
      "reader": "an-exterior-conformal-coordinate.html",
      "source": "src/an-exterior-conformal-coordinate.md",
      "source_sha256": "37dcce6f4812258fd3f1d95bcc114a0a08fd4961ba9c30f4a1a1076b8769c8a9",
      "status": "author_self_checked"
    },
    {
      "id": "faber-polynomials-and-a-positive-kernel",
      "title": "Faber polynomials and a positive kernel",
      "reader": "faber-polynomials-and-a-positive-kernel.html",
      "source": "src/faber-polynomials-and-a-positive-kernel.md",
      "source_sha256": "f091a92dc64b73efaf6a69b1e3c1552d9666936bd1635f03f1c93fcf1409a658",
      "status": "author_self_checked"
    },
    {
      "id": "the-bound-on-a-smooth-convex-domain",
      "title": "The bound on a smooth convex domain",
      "reader": "the-bound-on-a-smooth-convex-domain.html",
      "source": "src/the-bound-on-a-smooth-convex-domain.md",
      "source_sha256": "206cf6c4eb36acbf5b9e702a5d9fcf404c38e0fb81b4824d11cc0c18426cf31a",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "linear algebra over C: tensor products of matrices, Hermitian and positive semidefinite matrices, the Hilbert-Schmidt norm, singular vectors",
    "complex analysis as in the core course Complex Analysis (Lebl's Guide to Cultivating Complex Analysis, cited by number at each use): Cauchy's integral formula and estimates, Laurent series, the residue theorem, Rouche's theorem, the open mapping theorem, the Riemann mapping theorem, harmonic conjugates and the reflection principle for harmonic functions"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}