Page 156 - Textos de Matemática Vol. 44

146 H. NAKAZATO, A. KOVACˇEC, N. BEBIANO AND J. DA PROVIDEˆNCIA
the boundary of W2Π(diag(1, i, 0)) as a typical example of (1.1). Only few quan- titative properties are known for W3Π(D). We wish to provide a starting point for the quantitative study of such a range. We focus our study on the range
(1.3) W3Π(D0), where D0 = diag(ω, ω2, 1) and ω = (−1 + i√3)/2
is an imaginary cubic root of 1, as an example of (1.2) satisfying a typical symmetry. A method to produce a numerical approximation of such a range has already been given in [10]. Numerical approximation is useful to test the candidates of the boundary of the set (1.3). Our main theorem is the following.
Theorem 1.1. The set W3Π(D0) of products of diagonal entries of the unitary similarity orbit of the diagonal matrix with the cubic roots of 1 is star-shaped with 0 as a star-center. This set contains the parametrized curve
2eiθ + e−2iθ (1.4) [0, 2π/3] ∋ θ → x(θ) + iy(θ) = ( 3 )3,
where x(.), y(.) are real valued functions.
We shall prove this theorem in the next section. We present in Figure 1 a numerically produced proxy of the range (1.3). Quite possibly the boundary of this range is actually the curve (1.4). But this is not easy to establish.
0.1
0.05
0
-0.05
-0.1 0 0.2 0.4 0.6 0.8 1 Figure 1