Pencils of real symmetric matrices and real algebraic curves (Q921082)

In a real projective plane let \(\phi\) (x,y,z) be a homogeneous equation of degree n for a curve \(\Gamma\). Suppose (a) \(\phi (0,0,1)=1\); (b) every line passing through (0,0,1) intersects \(\Gamma\) in n real points (counting multiplications). The author shows that there are real symmetric matrices A and B of order n such that \(\phi (x,y,z)=\det (xA+yB+zI)\) if the curve \(\Gamma\) is rational or if every irreducible component of \(\Gamma\) is rational, but for \(n\geq 3\) not for general \(\Gamma\).