Critical values for higher rank numerical ranges associated with roulette curves (Q448359)

The critical value is determined for the higher rank numerical ranges of matrices associated with a~parameter of a~roulette curve (one of well known rational curves having circular symmetry), for which the higher rank numerical range is a regular polygon for every parameter less than or equal to the critical value. Let \(M_n\) be the algebra of \(n\times n\) complex matrices. For a~positive integer \(k \leq n\), the rank-\(k\)-numerical range of \(A \in M_n\) is defined as \(\Lambda_k(A)=\{\lambda\in \mathbb C\mid PAP=\lambda P\) for some rank \(k\) orthogonal projection~\(P\}\). The relationship between the asymptotic behavior of the one-parameter roulette curves and the change of the shape of \(\Lambda_k(A)\) as \(a\to 1+0\) is studied. The maximum value of \(a\) (the parameter of the roulette curve) is examined for which \(\Lambda_k(A)\) becomes a~regular polygon. A~result is obtained showing that the half turning angle~\(\beta\) at a~sharp point of the roulette curve is decreasing as the parameter \(\alpha\) increases.