Daniel Plaumann

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.

Determinantal representations of hyperbolic plane curves: An elementary approach

In 2007, Helton and Vinnikov proved that every hyperbolic plane curve has a definite real symmetric determinantal representation. By allowing for Hermitian matrices instead, we are able to give a new proof that relies only on the basic intersection theory of plane curves. We show that a matrix of linear forms is definite if and only if its co-maximal minors interlace its determinant and extend a classical construction of determinantal representations of Dixon from 1902. Like the Helton-Vinnikov theorem, this implies that every hyperbolic region in the plane is defined by a linear matrix inequality.

Hyperbolic polynomials, interlacers, and sums of squares

Hyperbolic polynomials are real polynomials whose real hypersurfaces are maximally nested ovaloids, the innermost of which is convex. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential equations. Here we investigate the special connection between a hyperbolic polynomial and the set of polynomials that interlace it. This set of interlacers is a convex cone, which we write as a linear slice of the cone of nonnegative polynomials. In particular, this allows us to realize any hyperbolicity cone as a slice of the cone of nonnegative polynomials. Using a sums of squares relaxation, we then approximate a hyperbolicity cone by the projection of a spectrahedron. A multiaffine example coming from the Vámos matroid shows that this relaxation is not always exact. Using this theory, we characterize the real stable multiaffine polynomials that have a definite determinantal representation and construct one when it exists.

Computing Linear Matrix Representations of Helton-Vinnikov Curves

Helton and Vinnikov showed that every rigidly convex curve in the real plane bounds a spectrahedron. This leads to the computational problem of explicitly producing a symmetric (positive definite) linear determinantal representation for a given curve. We study three approaches to this problem: an algebraic approach via solving polynomial equations, a geometric approach via contact curves, and an analytic approach via theta functions. These are explained, compared, and tested experimentally for low degree instances.

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 semidefinitely representable sets. Part of the interest in spectrahedra and semidefinitely representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming, such as 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 semidefinitely representable sets. Lasserre proposed a moment matrix method to construct semidefinite representations for certain sets. Our main result is that this method can work only if all faces of the considered set are exposed. This necessary condition complements sufficient conditions recently proved by Lasserre, Helton, and Nie.

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.

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.

Determinantal representations of hyperbolic curves via polynomial homotopy continuation

A smooth curve in the real projective plane is hyperbolic if its ovals are maximally nested. By the Helton-Vinnikov Theorem, any such curve admits a definite symmetric determinantal representation. We use polynomial homotopy continuation to compute such representations numerically. Our method works by lifting paths from the space of hyperbolic polynomials to a branched cover in the space of pairs of symmetric matrices.

A relative Grace Theorem for complex polynomials

We study the pullback of the apolarity invariant of complex polynomials in one variable under a polynomial map on the complex plane. As a consequence, we obtain variations of the classical results of Grace and Walsh in which the unit disk, or a circular domain, is replaced by its image under the given polynomial map.

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 Putinars 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.