Pencils of real symmetric matrices and real algebraic curves. (English) Zbl 0709.15009

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 \(\Gamma\).
Reviewer: G.P.Barker


15A22 Matrix pencils
14H45 Special algebraic curves and curves of low genus
15B57 Hermitian, skew-Hermitian, and related matrices
Full Text: DOI


[1] Fiedler, M., Geometry of the numerical range of matrices, Linear Algebra Appl., 37, 81-96 (1981) · Zbl 0452.15024
[2] Obreschkoff, N., Verteilung und Berechnung der Nullstellen reeller Polynome (1963), DVW: DVW Berlin · Zbl 0156.28202
[3] V. Vilhelm, Private communication.; V. Vilhelm, Private communication.
[4] Waerden, B. L.v.d., Moderne Algebra I (1937), Springer-Verlag: Springer-Verlag Berlin · JFM 63.0082.06
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