Hermitian Determinantal Representations of Hyperbolic Curves
نویسندگان
چکیده
(1) f = det(xM1 + yM2 + zM3), where M1,M2,M3 are Hermitian d × d matrices. The representation is definite if there is a point e ∈ R for which the matrix e1M1+e2M2+e3M3 is positive definite. This imposes an immediate condition on the projective curve VC(f). Because the eigenvalues of a Hermitian matrix are real, every real line passing through e meets this hypersurface in only real points. A polynomial with this property is called hyperbolic (with respect to e). Hyperbolicity is reflected in the topology of the real points VR(f). When the curve VC(f) is smooth, f is hyperbolic if and only if VR(f) consists of bd2c nested ovals, and a pseudo-line if d is odd.
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