421 research outputs found
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
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
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
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