arXiv subjects

Daniel Plaumann

Publications and source records attributed to Daniel Plaumann.

25 records · Page 2Linked to original sources

arXiv2012

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.

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arXiv2011

Determinantal Representations and the Hermite Matrix

We consider the problem of writing real polynomials as determinants of symmetric linear matrix polynomials. This problem of algebraic geometry, whose roots go back to the nineteenth century, has recently received new attention from the viewpoint of convex optimization. We relate the question to sums of squares decompositions of a certain Hermite matrix. If some power of a polynomial admits a definite determinantal representation, then its Hermite matrix is a sum of squares. Conversely, we show how a determinantal representation can sometimes be constructed from a sums-of-squares decomposition of the Hermite matrix. We finally show that definite determinantal representations always exist, if one allows for denominators.

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arXiv2011

Positivity of continuous piecewise polynomials

Real algebraic geometry provides certificates for the positivity of polynomials on semi-algebraic sets by expressing them as a suitable combination of sums of squares and the defining inequalitites. We show how Putinar's theorem for strictly positive polynomials on compact sets can be applied in the case of strictly positive piecewise polynomials on a simplicial complex. In the 1-dimensional case, we improve this result to cover all non-negative piecewise polynomials and give explicit degree bounds.

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arXiv2011

Quartic Curves and Their Bitangents

A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. These correspond to Cayley octads and Steiner complexes respectively. We present exact algorithms for computing these objects from the 28 bitangents. This expresses Vinnikov quartics as spectrahedra and positive quartics as Gram matrices. We explore the geometry of Gram spectrahedra and we find equations for the variety of Cayley octads. Interwoven is an exposition of much of the 19th century theory of plane quartics.

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arXiv2010

The ring of bounded polynomials on a semi-algebraic set

Let V be a normal affine variety over the real numbers R, and let S be a semi-algebraic subset of V(R). We study the subring B(S) of the coordinate ring of V consisting of the polynomials that are bounded on S. We introduce the notion of S-compatible completions of V, and we prove the existence of such completions when V is of dimension at most 2 or S=V(R). An S-compatible completion X of V yields an isomorphism of B(S) with the ring of regular functions on some (concretely specified) open subvariety of X. We prove that B(S) is a finitely generated R-algebra if S is open and of dimension at most 2, and we show that this result becomes false in higher dimensions.

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arXiv2009

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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arXiv2009

Sums of squares on reducible real curves

We ask whether every polynomial function that is non-negative on a real algebraic curve can be expressed as a sum of squares in the coordinate ring. Scheiderer has classified all irreducible curves for which this is the case. For reducible curves, we show how the answer depends on the configuration of the irreducible components and give complete necessary and sufficient conditions. We also prove partial results in the more general case of finitely generated preorderings and discuss applications to the moment problem for semialgebraic sets.

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