Determinantal representations of hyperbolic plane curves: An elementary approach

Abstract

If a real symmetric matrix of linear forms is positive definite at some point, then its determinant defines a hyperbolic hypersurface. In 2007, Helton and Vinnikov proved a converse in three variables, namely that every hyperbolic curve in the projective plane has a definite real symmetric determinantal representation. The goal of this paper is to give a more concrete proof of a slightly weaker statement. Here we show that every hyperbolic plane curve has a definite determinantal representation with Hermitian matrices. We do this by relating the definiteness of a matrix to the real topology of its minors and extending a construction of Dixon from 1902. Like the Helton-Vinnikov theorem, this implies that every hyperbolic region in the plane is defined by a linear matrix inequality. arXiv:1207.7047v2 [math.AG]

Spectrahedral representations of plane hyperbolic curves

“…Interlacers of f appear naturally in the context of determinantal representations of f (see [10,11]). For example, if f = det(xA + yB + zC) is a real symmetric and definite determinantal representation of f , then every principal (d−1)×(d−1) minor of xA+yB +zC is an interlacer of f (see [11,Thm. 3.3]).…”

“…with M (e) = I m . The determinant map taking a monic symmetric real linear matrix pencil of size m × m to its determinant is proper, hence its image is closed (see for example [11,Lemma 3.4]). If g is a strict interlacer of f , the pair (f, g) is in the closure of the set of pairs ( f , g), where f is hyperbolic with respect to e, V( f ) is smooth, and g is a strict interlacer of f satisfying the genericity assumptions (G1)-(G3).…”

Determinantal representations of invariant hyperbolic plane curves

“…The eigenvalues of this restriction are 1, ω, and ω n−1 with associated eigenspaces Λ (1) 1 , Λ (ω) 1 , and Λ ω n−1 1 . Since each 2 × 2 minor of G lies in the ideal f , each entry of adj(G) will be divisible by f n−2 by Theorem 4.6 of [13] and Step 6 is valid. Let M = adj(G) f n−2 .…”

“…Lax conjectured, in the context of hyperbolic differential operators, that every hyperbolic polynomial possesses a definite determinantal representation with real symmetric matrices [11]. Helton and Vinnikov [9] proved this conjecture in 2007, while Plaumann and Vinzant [13] gave a concrete construction for the Hermitian case in 2013. Figure 1: Hyperbolic dihedral invariant quintic f A associated to A = S(10 + 5i, 2 + 10i, 14, 16i, 12) in the hyperplane {t = 1} with connected component containing origin shaded (left).…”

Determinantal representations of hyperbolic forms via weighted shift matrices

“…Fiedler [10] and Helton-Vinnikov [15] also respectively provided an explicit formula for the real symmetric matrices H; K for the ternary form Fðt; x; yÞ. Recently, Plaumann and Vinzant [20] gave an elementary proof of the existence of hermitian matrices H; K for the hyperbolic form Fðt; x; yÞ, and hence the validity of the Fidler conjecture. An alternative proof of the Helton-Vinnikov theorem is given in [12].…”