Singular points of cyclic weighted shift matrices

Let S be an n-by-n   cyclic weighted shift matrix, and FS(t,x,y)=det(tI+xℜ(S)+yℑ(S))FS(t,x,y)=det(tI+xℜ(S)+yℑ(S)) be a ternary form associated with S  . We investigate the number of singular points of the curve FS(t,x,y)=0FS(t,x,y)=0, and show that the number of singular points of FS(t,x,y)=0FS(t,x,y)=0 associated with a cyclic weighted shift matrix whose weights are neither 1-periodic nor 2-periodic is less than or equal to n(n−3)/2n(n−3)/2. Furthermore, we verify the upper bound n(n−3)/2n(n−3)/2 is sharp for 4⩽n⩽74⩽n⩽7.