arXiv subjects

Thorsten Theobald

Publications and source records attributed to Thorsten Theobald.

At least 37 records · Page 2Linked to original sources

arXiv2019

Irreducible infeasible subsystems of semidefinite systems

Farkas' lemma for semidefinite programming characterizes semidefinite feasibility of linear matrix pencils in terms of an alternative spectrahedron. In the well-studied special case of linear programming, a theorem by Gleeson and Ryan states that the index sets of irreducible infeasible subsystems are exactly the supports of the vertices of the corresponding alternative polyhedron. We show that one direction of this theorem can be generalized to the nonlinear situation of extreme points of general spectrahedra. The reverse direction, however, is not true in general, which we show by means of counterexamples. On the positive side, an irreducible infeasible block subsystem is obtained whenever the extreme point has minimal block support. Motivated by results from sparse recovery, we provide a criterion for the uniqueness of solutions of semidefinite block systems.

math.OC↗

arXiv2018

Imaginary projections of polynomials

We introduce the imaginary projection of a multivariate polynomial $f \in \mathbb{C}[\mathbf{z}]$ as the projection of the variety of $f$ onto its imaginary part, $\mathcal{I}(f) \ = \ \{\text{Im}(\mathbf{z}) \, : \, \mathbf{z} \in \mathcal{V}(f) \}$. Since a polynomial $f$ is stable if and only if $\mathcal{I}(f) \cap \mathbb{R}_{>0}^n \ = \ \emptyset$, the notion offers a novel geometric view underlying stability questions of polynomials. We show that the connected components of the complement of the closure of the imaginary projections are convex, thus opening a central connection to the theory of amoebas and coamoebas. Building upon this, the paper establishes structural properties of the components of the complement, such as lower bounds on their maximal number, proves a complete classification of the imaginary projections of quadratic polynomials and characterizes the limit directions for polynomials of arbitrary degree.

math.AG↗

arXiv2018

Hyperbolicity cones and imaginary projections

Recently, the authors and de Wolff introduced the imaginary projection of a polynomial $f\in\mathbb{C}[\mathbf{z}]$ as the projection of the variety of $f$ onto its imaginary part, $\mathcal{I}(f) \ = \ \{\text{Im}(\mathbf{z}) \, : \, \mathbf{z} \in \mathcal{V}(f) \}$. Since a polynomial $f$ is stable if and only if $\mathcal{I}(f) \cap \mathbb{R}_{>0}^n \ = \ \emptyset$, the notion offers a novel geometric view underlying stability questions of polynomials. In this article, we study the relation between the imaginary projections and hyperbolicity cones, where the latter ones are only defined for homogeneous polynomials. Building upon this, for homogeneous polynomials we provide a tight upper bound for the number of components in the complement $\mathcal{I}(f)^{c}$ and thus for the number of hyperbolicity cones of $f$. And we show that for $n \ge 2$, a polynomial $f$ in $n$ variables can have an arbitrarily high number of strictly convex and bounded components in $\mathcal{I}(f)^{c}$.

math.AG↗

arXiv2018

Conic stability of polynomials

We introduce and study the notion of conic stability of multivariate complex polynomials in $\mathbb{C}[z_1,\ldots, z_n]$, which naturally generalizes the stability of multivariate polynomials. In particular, we generalize Borcea's and Brändén's multivariate version of the Hermite-Kakeya-Obreschkoff Theorem to the conic stability and provide a characterization in terms of a directional Wronskian. And we generalize a major criterion for stability of determinantal polynomials to stability with respect to the positive semidefinite cone.

math.CV↗

arXiv2017

Mixed Ehrhart polynomials

For lattice polytopes $P_1,\ldots, P_k \subseteq \mathbb{R}^d$, Bihan (2014) introduced the discrete mixed volume $\mathrm{DMV}(P_1,\dots,P_k)$ in analogy to the classical mixed volume. In this note we initiate the study of the associated mixed Ehrhart polynomial $\mathrm{ME}_{P_1,\dots,P_k}(n) = \mathrm{DMV}(nP_1,\dots,nP_k)$. We study properties of this polynomial and we give interpretations for some of its coefficients in terms of (discrete) mixed volumes. Bihan (2014) showed that the discrete mixed volume is always non-negative. Our investigations yield simpler proofs for certain special cases. We also introduce and study the associated mixed $h^*$-vector. We show that for large enough dilates $r P_1, \ldots, rP_k$ the corresponding mixed $h^*$-polynomial has only real roots and as a consequence the mixed $h^*$-vector becomes non-negative.

math.CO↗

arXiv2016

Sum of Squares Certificates for Containment of $\mathcal{H}$-polytopes in $\mathcal{V}$-polytopes

Given an $\mathcal{H}$-polytope $P$ and a $\mathcal{V}$-polytope $Q$, the decision problem whether $P$ is contained in $Q$ is co-NP-complete. This hardness remains if $P$ is restricted to be a standard cube and $Q$ is restricted to be the affine image of a cross polytope. While this hardness classification by Freund and Orlin dates back to 1985, for general dimension there seems to be only limited progress on that problem so far. Based on a formulation of the problem in terms of a bilinear feasibility problem, we study sum of squares certificates to decide the containment problem. These certificates can be computed by a semidefinite hierarchy. As a main result, we show that under mild and explicitly known preconditions the semidefinite hierarchy converges in finitely many steps. In particular, if $P$ is contained in a large $\mathcal{V}$-polytope $Q$ (in a well-defined sense), then containment is certified by the first step of the hierarchy.

math.CO↗

arXiv2015

Norms of Roots of Trinomials

The behavior of norms of roots of univariate trinomials $z^{s+t} + p z^t + q \in \mathbb{C}[z]$ for fixed support $A = \{0,t,s+t\} \subset \mathbb{N}$ with respect to the choice of coefficients $p,q \in \mathbb{C}$ is a classical late 19th and early 20th century problem. Although algebraically characterized by P.\ Bohl in 1908, the geometry and topology of the corresponding parameter space of coefficients had yet to be revealed. Assuming $s$ and $t$ to be coprime we provide such a characterization for the space of trinomials by reinterpreting the problem in terms of amoeba theory. The roots of given norm are parameterized in terms of a hypotrochoid curve along a $\mathbb{C}$-slice of the space of trinomials, with multiple roots of this norm appearing exactly on the singularities. As a main result, we show that the set of all trinomials with support $A$ and certain roots of identical norm, as well as its complement can be deformation retracted to the torus knot $K(s+t,s)$, and thus are connected but not simply connected. An exception is the case where the $t$-th smallest norm coincides with the $(t+1)$-st smallest norm. Here, the complement has a different topology since it has fundamental group $\mathbb{Z}^2$.

math.AG↗

arXiv2015

A Semidefinite Hierarchy for Containment of Spectrahedra

A spectrahedron is the positivity region of a linear matrix pencil and thus the feasible set of a semidefinite program. We propose and study a hierarchy of sufficient semidefinite conditions to certify the containment of a spectrahedron in another one. This approach comes from applying a moment relaxation to a suitable polynomial optimization formulation. The hierarchical criterion is stronger than a solitary semidefinite criterion discussed earlier by Helton, Klep, and McCullough as well as by the authors. Moreover, several exactness results for the solitary criterion can be brought forward to the hierarchical approach. The hierarchy also applies to the (equivalent) question of checking whether a map between matrix (sub-)spaces is positive. In this context, the solitary criterion checks whether the map is completely positive, and thus our results provide a hierarchy between positivity and complete positivity.

math.OC↗

arXiv2013

Approximating amoebas and coamoebas by sums of squares

Amoebas and coamoebas are the logarithmic images of algebraic varieties and the images of algebraic varieties under the arg-map, respectively. We present new techniques for computational problems on amoebas and coamoebas, thus establishing new connections between (co-)amoebas, semialgebraic and convex algebraic geometry and semidefinite programming. Our approach is based on formulating the membership problem in amoebas (respectively coamoebas) as a suitable real algebraic feasibility problem. Using the real Nullstellensatz, this allows to tackle the problem by sums of squares techniques and semidefinite programming. Our method yields polynomial identities as certificates of non-containment of a point in an amoeba or coamoeba. As the main theoretical result, we establish some degree bounds on the polynomial certificates. Moreover, we provide some actual computations of amoebas based on the sums of squares approach.

math.AG↗

arXiv2013

Amoebas of genus at most one

The amoeba of a Laurent polynomial $f \in \C[z_1^{\pm 1},\ldots,z_n^{\pm 1}]$ is the image of its zero set $\mathcal{V}(f)$ under the log-absolute-value map. Understanding the space of amoebas (i.e., the decomposition of the space of all polynomials, say, with given support or Newton polytope, with regard to the existing complement components) is a widely open problem. In this paper we investigate the class of polynomials $f$ whose Newton polytope $\New(f)$ is a simplex and whose support $A$ contains exactly one point in the interior of $\New(f)$. Amoebas of polynomials in this class may have at most one bounded complement component. We provide various results on the space of these amoebas. In particular, we give upper and lower bounds in terms of the coefficients of $f$ for the existence of this complement component and show that the upper bound becomes sharp under some extremal condition. We establish connections from our bounds to Purbhoo's lopsidedness criterion and to the theory of $A$-discriminants. Finally, we provide a complete classification of the space of amoebas for the case that the exponent of the inner monomial is the barycenter of the simplex Newton polytope. In particular, we show that the set of all polynomials with amoebas of genus 1 is path-connected in the corresponding space of amoebas, which proves a special case of the question on connectivity (for general Newton polytopes) stated by H. Rullgård.

math.AG↗

arXiv2013

Containment problems for polytopes and spectrahedra

We study the computational question whether a given polytope or spectrahedron $S_A$ (as given by the positive semidefiniteness region of a linear matrix pencil $A(x)$) is contained in another one $S_B$. First we classify the computational complexity, extending results on the polytope/polytope-case by Gritzmann and Klee to the polytope/spectrahedron-case. For various restricted containment problems, NP-hardness is shown. We then study in detail semidefinite conditions to certify containment, building upon work by Ben-Tal, Nemirovski and Helton, Klep, McCullough. In particular, we discuss variations of a sufficient semidefinite condition to certify containment of a spectrahedron in a spectrahedron. It is shown that these sufficient conditions even provide exact semidefinite characterizations for containment in several important cases, including containment of a spectrahedron in a polyhedron. Moreover, in the case of bounded $S_A$ the criteria will always succeed in certifying containment of some scaled spectrahedron $νS_A$ in $S_B$.

math.OC↗

arXiv2012

Exploiting symmetries in SDP-relaxations for polynomial optimization

In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semi definite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems, show how the degree principle for the symmetric group can be computationally exploited and also propose some methods to efficiently compute in the geometric quotient.

math.OC↗

arXiv2012

Determining a rotation of a tetrahedron from a projection

The following problem, arising from medical imaging, is addressed: Suppose that $T$ is a known tetrahedron in $\R^3$ with centroid at the origin. Also known is the orthogonal projection $U$ of the vertices of the image $ϕT$ of $T$ under an unknown rotation $ϕ$ about the origin. Under what circumstances can $ϕ$ be determined from $T$ and $U$?

math.MG↗

arXiv2010

Projections of tropical varieties and their self-intersections

We study algebraic and combinatorial aspects of (classical) projections of $m$-dimensional tropical varieties onto $(m+1)$-dimensional planes. Building upon the work of Sturmfels, Tevelev, and Yu on tropical elimination as well as the work of the authors on projection-based tropical bases, we characterize algebraic properties of the relevant ideals and provide a characterization of the dual subdivision (as a subdivision of a fiber polytope). This dual subdivision naturally leads to the issue of self-intersections of a tropical variety under projections. For the case of curves, we provide some bounds for the (unweighted) number of self-intersections of projections onto the plane and give constructions with many self-intersections.

math.AG↗

arXiv2009

Combinatorics and Genus of Tropical Intersections and Ehrhart Theory

Let $g_1, ..., g_k$ be tropical polynomials in $n$ variables with Newton polytopes $P_1, ..., P_k$. We study combinatorial questions on the intersection of the tropical hypersurfaces defined by $g_1, ..., g_k$, such as the $f$-vector, the number of unbounded faces and (in case of a curve) the genus. Our point of departure is Vigeland's work who considered the special case $k=n-1$ and where all Newton polytopes are standard simplices. We generalize these results to arbitrary $k$ and arbitrary Newton polytopes $P_1, ..., P_k$. This provides new formulas for the number of faces and the genus in terms of mixed volumes. By establishing some aspects of a mixed version of Ehrhart theory we show that the genus of a tropical intersection curve equals the genus of a toric intersection curve corresponding to the same Newton polytopes.

math.CO↗

arXiv2009

Mixed Volume Techniques for Embeddings of Laman Graphs

Determining the number of embeddings of Laman graph frameworks is an open problem which corresponds to understanding the solutions of the resulting systems of equations. In this paper we investigate the bounds which can be obtained from the viewpoint of Bernstein's Theorem. The focus of the paper is to provide the methods to study the mixed volume of suitable systems of polynomial equations obtained from the edge length constraints. While in most cases the resulting bounds are weaker than the best known bounds on the number of embeddings, for some classes of graphs the bounds are tight.

math.CO↗

arXiv2007

Tropical bases by regular projections

We consider the tropical variety $\mathcal{T}(I)$ of a prime ideal $I$ generated by the polynomials $f_1, ..., f_r$ and revisit the regular projection technique introduced by Bieri and Groves from a computational point of view. In particular, we show that $I$ has a short tropical basis of cardinality at most $r + \codim I + 1$ at the price of increased degrees, and we provide a computational description of these bases.

math.AG↗

arXiv2007

Enumerating the Nash equilibria of rank 1-games

A bimatrix game $(A,B)$ is called a game of rank $k$ if the rank of the matrix $A+B$ is at most $k$. We consider the problem of enumerating the Nash equilibria in (non-degenerate) games of rank 1. In particular, we show that even for games of rank 1 not all equilibria can be reached by a Lemke-Howson path and present a parametric simplex-type algorithm for enumerating all Nash equilibria of a non-degenerate game of rank 1.

cs.GT↗