Spectrahedral representations of plane hyperbolic curves

Abstract

We describe a new method for constructing a spectrahedral representation of the hyperbolicity region of a hyperbolic curve in the real projective plane. As a consequence, we show that if the curve is smooth and defined over the rational numbers, then there is a spectrahedral representation with rational matrices. This generalizes a classical construction for determinantal representations of plane curves due to Dixon and relies on the special properties of real hyperbolic curves that interlace the given curve.It coincides with the hyperbolicity cone C(f, e) of f = det M in direction e, that is the closure of the connected component of {a ∈ R 3 : f (a) = 0} containing e. This is a convex cone in R 3 , whose image in P 2 is the region enclosed by the convex innermost oval of the curve (see Figure 1). A triple of real symmetric matrices A, B, C is a spectrahedral representation of C(f, e) if M = xA + yB + zC satisfies C(f, e) = S(M ).

Certifying Polynomial Nonnegativity via Hyperbolic Optimization

“…The argument we used to show that every ternary hyperbolic polynomial is SOS-hyperbolic actually generalizes to give a projected spectrahedral description of the cone of interlacers of any ternary hyperbolic polynomial. This was first observed by Kummer, Naldi, and Plaumann [KNP18] via a seemingly quite different proof. Proof.…”

“…For a fixed hyperbolic polynomial p ∈ H n,d (e) the cone Int e (p) of polynomials q that interlace p with respect to e is a convex cone. In [KPV15,KNP18] it is shown that the cone of interlacers of p ∈ H n,d (e) can be described by Much of the development of Sections 3 and 6 could have been presented from the point of view of interlacers. This is because D u p(x) ∈ Int e (p) if, and only if, u ∈ Λ + (p, e), so the hyperbolicity cone is a section of the cone of interlacers [KPV15].…”

Spectral linear matrix inequalities

“…The first claim is clear since A is in the interior of the hyperbolicity cones of both P d and P d+1 . In order to prove the second claim, we proceed as in[KNP19, p. 261]. By Lemma 6.1 we have thatΦ(X) • M (X) = w • P d+1 (X)where w is the vector whose first N entries are 1 and all other entries are 0.…”