×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.