Livsic-type Determinantal Representations and Hyperbolicity

Abstract

Hyperbolic homogeneous polynomials with real coefficients, i.e., hyperbolic real projective hypersurfaces, and their determinantal representations, play a key role in the emerging field of convex algebraic geometry. In this paper we consider a natural notion of hyperbolicity for a real subvariety $X\subset {\mathbb{P}}^d$ of an arbitrary codimension $\ell$ with respect to a real $\ell -1$-dimensional linear subspace $V\subset {\mathbb{P}}^d$ and study its basic properties. We also consider a special kind of determinantal representations that we call Livsic-type and a nice subclass of these that we call \vr{}. Much like in the case of hypersurfaces ($\ell =1$), the existence of a definite Hermitian \vr{} Livsic-type determinantal representation implies hyperbolicity. We show that every curve admits a \vr{} Livsic-type determinantal representation. Our basic tools are Cauchy kernels for line bundles and the notion of the Bezoutian for two meromorphic functions on a compact Riemann surface that we introduce. We then proceed to show that every real curve in ${\mathbb{P}}^d$ hyperbolic with respect to some real $d-2$-dimensional linear subspace admits a definite Hermitian, or even real symmetric, \vr{} Livsic-type determinantal representation.

References in corpus (3)

• The Lax conjecture is true
• The resultant on compact Riemann surfaces
• Rational transformations of algebraic curves and elimination theory

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