arXiv subjects

Rainer Sinn

Publications and source records attributed to Rainer Sinn.

43 records · Page 3Linked to original sources

arXiv2015

Real Rank with Respect to Varieties

We study the real rank of points with respect to a real variety $X$. This is a generalization of various tensor ranks, where $X$ is in a specific family of real varieties like Veronese or Segre varieties. The maximal real rank can be bounded in terms of the codimension of $X$ only. We show constructively that there exist varieties $X$ for which this bound is tight. The same varieties provide examples where a previous bound of Blekherman-Teitler on the maximal $X$-rank is tight. We also give examples of varieties $X$ for which the gap between maximal complex and the maximal real rank is arbitrarily large. To facilitate our constructions we prove a conjecture of Reznick on the maximal real symmetric rank of symmetric bivariate tensors. Finally we study the geometry of the set of points of maximal real rank in the case of real plane curves.

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arXiv2015

Extreme Rays of the Hankel Spectrahedra for Ternary Forms

Hankel spectrahedra are the dual convex cones to the cone of sums of squares of real polynomials, and we study them from the point of view of convex algebraic geometry. We show that the Zariski closure of the union of all extreme rays of Hankel spectrahedra for ternary forms is an irreducible variety of codimension 10. It is the variety of all Hankel (or middle Catalecticant) matrices of corank at least 4. We explicitly construct a rational extreme ray of maximal rank using the Cayley-Bacharach Theorem for plane curves. We work out the rank stratification of the semi-algebraic set of extreme rays of Hankel spectrahedra in the first three nontrivial cases d = 3, 4, 5. Dually, we get a characterisation of the algebraic boundary of the cone of sums of squares via projective duality theory, extending previous work of Blekherman, Hauenstein, Ottem, Ranestad, and Sturmfels.

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arXiv2015

Generic Spectrahedral Shadows

Spectrahedral shadows are projections of linear sections of the cone of positive semidefinite matrices. We characterize the polynomials that vanish on the boundaries of these convex sets when both the section and the projection are generic.

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arXiv2015

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.

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arXiv2014

Algebraic Boundaries of Convex Semi-algebraic Sets

We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalize the correspondence of facets of a polytope to the vertices of the dual polytope to general semi-algebraic convex bodies. In the general setup, exceptional families of extreme points might exist and we characterize them semi-algebraically. We also give an algorithm to compute a complete list of exceptional families, given the algebraic boundary of the dual convex set.

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arXiv2011

The Algebraic Boundary of SO(2)-Orbitopes

Let $X\subset\A^{2r}$ be a real curve embedded into an even-dimensional affine space. In the main result of this paper, 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 as semi-algebraic sets. In the case of 4-dimensional SO(2)-orbitopes, we find all irreducible components of their algebraic boundary.

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arXiv2009

A Note on the Convex Hull of Finitely Many Projections of Spectrahedra

A spectrahedron is a set defined by a linear matrix inequality. A projection of a spectrahedron is often called a semidefinitely representable set. We show that the convex hull of a finite union of such projections is again a projection of a spectrahedron. This improves upon the result of Helton and Nie, who prove the same result in the case of bounded sets.

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