Singular points of the algebraic curves associated to unitary bordering matrices

Let A be an n x n complex matrix. A ternary form associated to A is defined as the homogeneous polynomial F-A(t, x, y) = det(tI(n) + xR(A) + yJ(A)). We prove, for a unitary boarding matrix A, the ternary form F-A(t, x, y) is strongly hyperbolic and the algebraic curve F-A(t, x, y) = 0 has no real singular points. As a consequence, we obtain that the higher rank numerical range of a unitary boarding matrix is strictly convex. (C) 2016 Elsevier Inc. All rights reserved.