The boundary of the Q-numerical range of some Toeplitz nilpotent matrix (Q1685901)

Let \(A\) be an \(n\times n\) complex matrix. The usual numerical range \(W(A)\) is defined to be set of all \(\xi^{\ast}A\xi\) (\(\xi\in\mathbb{C}^{n}\), \(\left\| \xi\right\| =1\)). For \(\left| q\right| \leq1\) the \(q\)-numerical range \(W_{q}(A)\) consists of all \(\eta^{\ast}A\xi\) (\(\eta,\xi\in\mathbb{C} ^{n}\), \(\left\| \eta\right\| =\left\| \xi\right\| =1\) and \(\eta^{\ast}\xi=q\)) (see, for example [\textit{C.-K. Li} and \textit{H. Nakazato}, Linear Multilinear Algebra 43, No. 4, 385--409 (1998; Zbl 0899.15017)]). The authors are interested in computing the boundary of \(W_{q}(A)\). This has been done by other authors for some typical \(3\times3\) matrices and it is believed that the boundary for a generic \(3\times3\) matrix can be described by a polynomial of degree \(24\). In the present paper the authors consider a particular \(4\times4\) nilpotent matrix \(N\) with rows \(0~1~0~1;~0~0~1~0;~0~0~0~1;~0~0~0~0\). Taking \(q=1599/1601\) (part of a Pythagorean triple so that the polynomial is rational) they construct a homogeneous polynomial \(g(x,y)\) with \(253\) terms and of degree \(40\) such that the boundary points \(x+iy\) of \(W_{q}(N)\) are the roots of \(g(x,y)\). Similarly they construct a homogeneous rational polynomial \(L_{0,N}(X,Y,Z)\) with \(135\) terms and of degree \(12\) whose roots are determined by the boundary points \(X+iY=\xi^{\ast}N\xi\) of \(W(N)\) and \(Z=\xi^{\ast}N^{\ast}N\xi\). Some Mathematica code to compute the boundary points numerically is included.