Singular points of the algebraic curves associated to unitary bordering matrices

Let A   be an n×nn×n complex matrix. A ternary form associated to A   is defined as the homogeneous polynomial FA(t,x,y)=det⁡(tIn+xℜ(A)+yℑ(A))FA(t,x,y)=det⁡(tIn+xℜ(A)+yℑ(A)). We prove, for a unitary boarding matrix A  , the ternary form FA(t,x,y)FA(t,x,y) is strongly hyperbolic and the algebraic curve FA(t,x,y)=0FA(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.