The Clouatre-Ostermann-Ransford conjecture is proved using positivity, an operator-valued Herglotz theorem, and the Lorist-Schwenninger perturbation lemma.
A solution to Crouzeix's conjecture
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abstract
We provide a proof of Crouzeix's conjecture, which combines the tools developed previously for weaker estimates with a simple perturbation lemma for $2$-dilations. Applying the lemma to the iterates $f^{n}$ in the double-layer potential representation yields the conjectured bound.
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2026 4representative citing papers
For every 3 by 3 complex matrix A, every matrix-valued polynomial F satisfies ||F(A)|| <= 2 max_{z in W(A)} ||F(z)||, so W(A) is a complete 2-spectral set.
For every nonzero q with |q|<=1, the sharp spectral constant of the scaled q-numerical range is max{1, 2|q|/(1+sqrt(1-|q|^2))}.
Autonomous LLM agents in the Station environment found new finite-field Kakeya sets, 604-point kissing configurations in dimension 11, new bounds on three further optimization problems, and two new infinite families for Book Ramsey numbers.
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On the abstract approach to spectral constants: a proof of the Clou\^atre--Ostermann--Ransford conjecture
The Clouatre-Ostermann-Ransford conjecture is proved using positivity, an operator-valued Herglotz theorem, and the Lorist-Schwenninger perturbation lemma.
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Square Functions and the Complete Crouzeix Conjecture in Dimension Three
For every 3 by 3 complex matrix A, every matrix-valued polynomial F satisfies ||F(A)|| <= 2 max_{z in W(A)} ||F(z)||, so W(A) is a complete 2-spectral set.
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Sharp spectral constants for scaled $q$-numerical ranges
For every nonzero q with |q|<=1, the sharp spectral constant of the scaled q-numerical range is max{1, 2|q|/(1+sqrt(1-|q|^2))}.
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Autonomous Mathematical Discovery in an Open-World Multi-Agent Environment
Autonomous LLM agents in the Station environment found new finite-field Kakeya sets, 604-point kissing configurations in dimension 11, new bounds on three further optimization problems, and two new infinite families for Book Ramsey numbers.