Rainer Sinn

Computing Hermitian determinantal representations of hyperbolic curves

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.

Do Sums of Squares Dream of Free Resolutions

We associate to a real projective variety

Extreme rays of Hankel spectrahedra for ternary forms

X

Real rank with respect to varieties

two convex cones which are fundamental in real algebraic geometry: the cone

Low-Rank Sum-of-Squares Representations on Varieties of Minimal Degree

P_X

Gram Spectrahedra.

of quadratic forms nonnegative on

Algebraic Boundaries of Convex Semi-Algebraic Sets

X

Generic Completion Rank and Maximum Likelihood Threshold of Graphs

, and the cone

Extreme Rays of the Hankel Spectrahedra for Ternary Forms

\Sigma_X

Typical and Generic Ranks in Matrix Completion

of sums of squares of linear forms. The dual cone