Toric completions and bounded functions on real algebraic varieties

Given a semi-algebraic set S, we study compactifications of S that arise from embeddings into complete toric varieties. This makes it possible to describe the asymptotic growth of polynomial functions on S in terms of combinatorial data. We extend our earlier work to compute the ring of bounded functions in this setting and discuss applications to positive polynomials and the moment problem. Complete results are obtained in special cases, like sets defined by binomial inequalities. We also show that the wild behaviour of certain examples constructed by Krug and by Mondal-Netzer cannot occur in a toric setting.Comment: 19 pages; minor updates and correction

Exposed faces of semidefinitely representable sets

A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinite representable sets. Part of the interest in spectrahedra and semidefinite representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming, like one can do on polyhedra by linear programming. It is known that every face of a spectrahedron is exposed. This is also true in the general context of rigidly convex sets. We study the same question for semidefinite representable sets. Lasserre proposed a moment matrix method to construct semidefinite representations for certain sets. Our main result is that this method can only work if all faces of the considered set are exposed. This necessary condition complements sufficient conditions recently proved by Lasserre, Helton and Nie

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