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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

##### MSC:
 15A22 Matrix pencils 14H45 Special algebraic curves and curves of low genus 15B57 Hermitian, skew-Hermitian, and related matrices
##### Keywords:
real projective plane; real symmetric matrices
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##### References:
 [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 Berlin · Zbl 0156.28202 [3] V. Vilhelm, Private communication. [4] Waerden, B.L.v.d., Moderne algebra I, (1937), Springer-Verlag Berlin
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