71 research outputs found
Computing Linear Matrix Representations of Helton-Vinnikov Curves
Helton and Vinnikov showed that every rigidly convex curve in the real plane
bounds a spectrahedron. This leads to the computational problem of explicitly
producing a symmetric (positive definite) linear determinantal representation
for a given curve. We study three approaches to this problem: an algebraic
approach via solving polynomial equations, a geometric approach via contact
curves, and an analytic approach via theta functions. These are explained,
compared, and tested experimentally for low degree instances.Comment: 19 pages, 3 figures, minor revisions; Mathematical Methods in
Systems, Optimization and Control, Birkhauser, Base
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
Quartic Curves and Their Bitangents
A smooth quartic curve in the complex projective plane has 36 inequivalent
representations as a symmetric determinant of linear forms and 63
representations as a sum of three squares. These correspond to Cayley octads
and Steiner complexes respectively. We present exact algorithms for computing
these objects from the 28 bitangents. This expresses Vinnikov quartics as
spectrahedra and positive quartics as Gram matrices. We explore the geometry of
Gram spectrahedra and we find equations for the variety of Cayley octads.
Interwoven is an exposition of much of the 19th century theory of plane
quartics.Comment: 26 pages, 3 figures, added references, fixed theorems 4.3 and 7.8,
other minor change
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
Semidefinite Representation of the -Ellipse
The -ellipse is the plane algebraic curve consisting of all points whose
sum of distances from given points is a fixed number. The polynomial
equation defining the -ellipse has degree if is odd and degree
if is even. We express this polynomial equation as
the determinant of a symmetric matrix of linear polynomials. Our representation
extends to weighted -ellipses and -ellipsoids in arbitrary dimensions,
and it leads to new geometric applications of semidefinite programming.Comment: 16 pages, 5 figure
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