Computing Hermitian determinantal representations of hyperbolic curves
2015, Plaumann, Daniel, Sinn, Rainer, Speyer, David E., Vinzant, Cynthia
Every real hyperbolic form in three variables can be realized as the determinant of a linear net of Hermitian matrices containing a positive definite matrix. Such representations are an algebraic certificate for the hyperbolicity of the polynomial and their existence has been proved in several different ways. However, the resulting algorithms for computing determinantal representations are computationally intensive. In this note, we present an algorithm that reduces a large part of the problem to linear algebra and discuss its numerical implementation.
Algebraic boundaries of SO(2)-orbitopes
2013, Sinn, Rainer
Let X⊂A2r be a real curve embedded into an even-dimensional affine space. We characterise when the r th secant variety to X is an irreducible component of the algebraic boundary of the convex hull of the real points X(R) of X. This fact is then applied to 4 -dimensional SO(2) -orbitopes and to the so called Barvinok–Novik orbitopes to study when they are basic closed semi-algebraic sets. In the case of 4 -dimensional SO(2) -orbitopes, we find all irreducible components of their algebraic boundary.