Crouzeix-Conjecture has fallen

It happened again. A machine has solved a conjecture in linear algebra. it was a beautiful problem, now reduced to junk after letting Chat GPT run through it for 16 hours. Like for an other beautiful conjecture involving polynomials, the Sendov conjecture (Tao wrote about it recently), we see the AI cancer eat away beautiful problems. As the German article about the Creouzix conjecture reports, humans are still able to digest the proof and verify it. The cancer AI has become worse.

Take a polynomial p and a complex M x M matrix A. The conjecture claims that the maximal spectral norm of $p(A)$ is smaller or equal than twice the maximum of |p(z)| where z runs over the numerical range $W=\{ \overline{z} p(A) z/|z|^2, z \neq 0 \}$ of p(A). The spectral norm is the $l^2$ norm of the spectrum when seen as a vector in a complex Hilbert space. It is the sum $(\sum_{j=1}^M |p(\lambda_j)|^2)^{(1/2)}$ .

Lets look at the example of the 2 x 2 matrix $A=\left[ \begin{array}{cc} 3 & 1 \\ 2 & 4 I \\ \end{array} \right]$ . It has eigenvalues -2.3+3.7 i, 3.3+0.3 i. The numerical range W(A) of A is here an ellipse shown to the left. It is always a convex region containing the eigenvalues. Now lets take a polynomial like $p(z)=z^3-z$ . We can use this map to deform W conformally. This new region p(W) is shown to the right. The topic of conformal maps was probably the reason why the Chinese surgeon who prompted the AI was interested in the conjecture. The l2 norm of p(A) is here about 54.5. The maximal numerical range is 71.

Here is some code to experiment with.

M=2;A={{3,1},{2,4I}}; p[z_]:=z^3-z;  (*   A=Table[R,{M},{M}];  *)
NumericalRange[v_]:=Transpose[Conjugate[v]].A.v/(Conjugate[v].v); 
ComplexToPlane[z_]:={Re[z],Im[z]}; 
R:=2Random[]-1+2I Random[]-I;  RandomVector:=Table[R,{M}];
S=Table[ComplexToPlane[p[NumericalRange[RandomVector]]],{100000}];
L=Map[p,Eigenvalues[1.0*A]]; 
Print["Maximal eigenvalue p(z) ", Max[Abs[1.0*L]]];
Print["Maximal numerical range ", Max[Abs[1.0*S]]];

Author: oliverknill