{"id":"openai-math/A-direct-proof-of-the-complete-Crouzeix-inequality","pdf":"https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026","tags":["matrix-analysis","numerical-range","polynomial-inequalities"],"score":0,"title":"A direct proof of the complete Crouzeix inequality","domain":"Functional analysis","family":"325","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.FA","summary":"We give a direct proof of the sharp constant-two numerical-range inequality for matrix-valued polynomials in all finite base and coefficient dimensions. This resolves the complete Crouzeix conjecture in its matrix formulation, including matrices whose numerical ranges are points or line segments.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-26","reactions":{},"formalized":true,"family_title":"The complete Crouzeix conjecture"}