arXiv subjects
Publications and source records attributed to Mario Kummer.
A totally real theta characteristic of a real curve is a theta characteristic which is linearly equivalent to a sum of only real points. These are closely related to the facets of the convex hull of the canonical embedding of the curve.
We introduce the separating semigroup of a real algebraic curve of dividing type. The elements of this semigroup record the possible degrees of the covering maps obtained by restricting separating morphisms to the real part of the curve. We also introduce the hyperbolic semigroup which consists of elements of the separating semigroup arising from morphisms which are compositions of a linear projection with an embedding of the curve to some projective space. We completely determine both semigroups in the case of maximal curves. We also prove that any embedding of a real curve to projective space of sufficiently high degree is hyperbolic. Using these semigroups we show that the hyperbolicity locus of an embedded curve is in general not connected.
In this paper we study flat deformations of real subschemes of $\mathbb{P}^n$, hyperbolic with respect to a fixed linear subspace, i.e. admitting a finite surjective and real fibered linear projection. We show that the subset of the corresponding Hilbert scheme consisting of such subschemes is closed and connected in the classical topology. Every smooth variety in this set lies in the interior of this set. Furthermore, we provide sufficient conditions for a hyperbolic subscheme to admit a flat deformation to a smooth hyperbolic subscheme. This leads to new examples of smooth hyperbolic varieties.
We present a new perspective on the Schottky problem that links numerical computing with tropical geometry. The task is to decide whether a symmetric matrix defines a Jacobian, and, if so, to compute the curve and its canonical embedding. We offer solutions and their implementations in genus four, both classically and tropically. The locus of cographic matroids arises from tropicalizing the Schottky-Igusa modular form.
In this paper we define and study real fibered morphisms. Such morphisms arise in the study of real hyperbolic hypersurfaces in projective space and other hyperbolic varieties. We show that real fibered morphisms are intimately connected to Ulrich sheaves admitting positive definite symmetric bilinear forms.
We describe a new method for constructing a spectrahedral representation of the hyperbolicity region of a hyperbolic curve in the real projective plane. As a consequence, we show that if the curve is smooth and defined over the rational numbers, then there is a spectrahedral representation with rational matrices. This generalizes a classical construction for determinantal representations of plane curves due to Dixon and relies on the special properties of real hyperbolic curves that interlace the given curve.
A quadrature rule of a measure $μ$ on the real line represents a convex combination of finitely many evaluations at points, called nodes, that agrees with integration against $μ$ for all polynomials up to some fixed degree. In this paper, we present a bivariate polynomial whose roots parametrize the nodes of minimal quadrature rules for measures on the real line. We give two symmetric determinantal formulas for this polynomial, which translate the problem of finding the nodes to solving a generalized eigenvalue problem.
We present a computational study of smooth curves of degree six in the real projective plane. In the Rokhlin-Nikulin classification, there are 56 topological types, refined into 64 rigid isotopy classes. We developed software that determines the topological type of a given sextic and used it to compute empirical probability distributions on the various types. We list 64 explicit representatives with integer coefficients, and we perturb these to draw many samples from each class. This allows us to explore how many of the bitangents, inflection points and tensor eigenvectors are real. We also study the real tensor rank, the construction of quartic surfaces with prescribed topology, and the avoidance locus, which is the locus of all real lines that do not meet a given sextic. This is a union of up to 46 convex regions, bounded by the dual curve.
We give an elementary proof for the fact that an irreducible hyperbolic polynomial has only one pair of hyperbolicity cones.
It was observed by Bump et al. that Ehrhart polynomials in a special family exhibit properties similar to the Riemann ζ function. The construction was generalized by Matsui et al. to a larger family of reflexive polytopes coming from graphs. We prove several conjectures confirming when such polynomials have zeros on a certain line in the complex plane. Our main new method is to prove a stronger property called interlacing.
The geometry of the set of restrictions of rank-one tensors to some of their coordinates is studied. This gives insight into the problem of rank-one completion of partial tensors. Particular emphasis is put on the semialgebraic nature of the problem, which arises for real tensors with constraints on the parameters. The algebraic boundary of the completable region is described for tensors parametrized by probability distributions and where the number of observed entries equals the number of parameters. If the observations are on the diagonal of a tensor of format $d\times\dots\times d$, the complete semialgebraic description of the completable region is found.
A reciprocal linear space is the image of a linear space under coordinate-wise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and appear as part of the defining equations of the central path of a linear program. Their structure is controlled by an underlying matroid. This provides a large family of hyperbolic varieties, recently introduced by Shamovich and Vinnikov. Here we give a definite determinantal representation to the Chow form of a reciprocal linear space. One consequence is the existence of symmetric rank-one Ulrich sheaves on reciprocal linear spaces. Another is a representation of the entropic discriminant as a sum of squares. For generic linear spaces, the determinantal formulas obtained are closely related to the Laplacian of the complete graph and generalizations to simplicial matroids. This raises interesting questions about the combinatorics of hyperbolic varieties and connections with the positive Grassmannian.
Given an integral domain A we consider the set of all integral elements over A that can occur as an eigenvalue of a symmetric matrix over A. We give a sufficient criterion for being such an element. In the case where A is the ring of integers of an algebraic number field this sufficient criterion is also necessary and we show that the size of matrices grows linearly in the degree of the element. The latter result settles a questions raised by Bass, Estes and Guralnick.
In this paper we study the real rank of monomials and we give an upper bound for the real rank of all monomials. We show that the real and the complex ranks of a monomial coincide if and only if the least exponent is equal to one.
The specialized Vámos polynomial is a hyperbolic polynomial of degree four in four variables with the property that none of its powers admits a definite determinantal representation. We will use a heuristical method to prove that its hyperbolicity cone is a spectrahedron.
A spectrahedron is a set defined by a linear matrix inequality. Given a spectrahedron we are interested in the question of the smallest possible size $r$ of the matrices in the description by linear matrix inequalities. We show that for the $n$-dimensional unit ball $r$ is at least $\frac{n}{2}$. If $n=2^k+1$, then we actually have $r=n$. The same holds true for any compact convex set in $\mathbb{R}^n$ defined by a quadratic polynomial. Furthermore, we show that for a convex region in $\mathbb{R}^3$ whose algebraic boundary is smooth and defined by a cubic polynomial we have that $r$ is at least five. More precisely, we show that if $A,B,C$ are real symmetric matrices such that $f(x,y,z)=\det(I+A x+B y+C z)$ is a cubic polynomial, the surface in complex projective three-space with affine equation $f(x,y,z)=0$ is singular.
We show that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that qh has a definite determinantal representation. This is proved by considering sum-of-squares decompositions of certain bilinear forms called Bézoutians. Besides commutative algebra, the proof relies on results from real algebraic geometry.
Hyperbolic polynomials are real polynomials whose real hypersurfaces are nested ovaloids, the inner most 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 Vamos 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.