126 research outputs found
Determinantal representations of hyperbolic plane curves: An elementary approach
If a real symmetric matrix of linear forms is positive definite at some
point, then its determinant is a hyperbolic hypersurface. In 2007, Helton and
Vinnikov proved a converse in three variables, namely that every hyperbolic
plane curve 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 Helton and Vinnikov's theorem, this
implies that every hyperbolic region in the plane is defined by a linear matrix
inequality.Comment: 15 pages, 4 figures, minor revision
Livsic-type Determinantal Representations and Hyperbolicity
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 of an arbitrary codimension with respect to a real -dimensional linear subspace 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 (), 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
hyperbolic with respect to some real -dimensional linear
subspace admits a definite Hermitian, or even real symmetric, \vr{} Livsic-type
determinantal representation
LMI Representations of Convex Semialgebraic Sets and Determinantal Representations of Algebraic Hypersurfaces: Past, Present, and Future
10 years ago or so Bill Helton introduced me to some mathematical problems
arising from semidefinite programming. This paper is a partial account of what
was and what is happening with one of these problems, including many open
questions and some new results
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