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F (t, x, y) = det(tIn + x(T + T )/2 + y(T − T )/(2i)). Let ΓF be the algebraic curve of F (t, x, y), i.e., ΓF = {[(t, x, y)] ∈ CP : F (t, x, y) = 0}, where [(t, x, y)] is the equivalence class containing (t, x, y) ∈ C − (0, 0, 0) under the relation (t1, x1, y1) ∼ (t2, x2, y2) if (t2, x2, y2) = k(t1, x1, y1) for some nonzero complex number k. The dual curve Γ∧F of ΓF is defined by Γ∧F = {[(T,X, Y )] ∈ CP : T t+X x+ Y y = 0 is a tangent line of ΓF }.

Singular points of the ternary polynomials associated with 4-by-4 matrices