arXiv subjects

Timo de Wolff

Publications and source records attributed to Timo de Wolff.

42 records · Page 3Linked to original sources

arXiv2014

A Sharp Upper Bound for the Complexity of Labeled Oriented Trees

A labeled oriented graph (LOG) is an oriented graph with a labeling function from the edge set into the vertex set. The complexity of a LOG is the minimal cardinality of an initial set $S$ of vertices such that every vertex can be reached successively from $S$ only using edges with labels in $S$ or already visited vertices. We give a constructive proof of a conjecture by Rosebrock stating that for an interior reduced, connected LOG with $m$ vertices the complexity is at most $(m+1) / 2$ and show that this bound is sharp. Due to results of Howie labeled oriented trees (LOTs) yield crucial candidates for counterexamples of the Whitehead Conjecture stating that every subcomplex of an aspherical 2-complex is aspherical. We explicitly describe the structure of LOTs of maximal complexity $(m+1)/2$. We conclude that the 2-complexes associated to these LOTs are always aspherical excluding them from the list of possible counterexamples.

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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.

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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.

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arXiv2013

Low Dimensional Test Sets for Nonnegativity of Even Symmetric Forms

An important theorem by Timofte states that nonnegativity of real $n$-variate symmetric polynomials of degree $d$ can be decided at test sets given by all points with at most $\lfloor\frac{d}{2}\rfloor$ distinct components. However, if the degree is sufficiently larger than the number of variables, then the theorem obviously does not provide nontrivial information. Our approach is to look at $(m + 1)$-dimensional subspaces of even symmetric forms of degree 4d, at which nonnegativity can be checked at $(m - 1)$-points, i.e., points with at most $m - 1 \in \N$ distinct components, where $m$ is independent of the degree of the forms and better than Timofte's bound. Furthermore, for fixed $k \in \N$, we tackle problems concerning the maximum dimension of such subspaces, at which nonnegativity can be checked at all $k$-points, as well as the geometrical and topological structure of the set of all forms whose nonnegativity can be decided at all $k$-points.

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arXiv2012

Separating inequalities for nonnegative polynomials that are not sums of squares

Ternary sextics and quaternary quartics are the smallest cases where there exist nonnegative polynomials that are not sums of squares (SOS). A complete classification of the difference between these cones was given by G. Blekherman via analyzing the extreme rays of the corresponding dual cones. However, an exact computational approach in order to build separating extreme rays for nonnegative polynomials that are not sums of squares is a widely open problem. We provide a method substantially simplifying this computation for certain classes of polynomials on the boundary of the PSD cones. In particular, our method yields separating extreme rays for every nonnegative ternary sextic with at least seven zeros. As an application to further instances, we compute a rational certificate proving that the Motzkin polynomial is not SOS.

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arXiv2010

Polytopes with Special Simplices

For a polytope P a simplex S with vertex set V(S) is called a special simplex if every facet of P contains all but exactly one vertex of S. For such polytopes P with face complex F(P) containing a special simplex the subcomplex F(P) / V(S) of all faces not containing vertices of S is the boundary of a polytope Q - the basis polytope of P. If additionally the dimension of the affine basis space of F(P) / V(S) equals dim(Q), we call P meek; otherwise we call P wild. We give a full combinatorial classification and techniques for geometric construction of the class of meek polytopes with special simplices. We show that every wild polytope P' with special simplex can be constructed out of a particular meek one P by intersecting P with particular hyperplanes. It is non-trivial to find all these hyperplanes for an arbitrary basis polytope; we give an exact description for 2-basis polytopes. Furthermore we show that the f-vector of each wild polytope with special simplex is component wise bounded above by the f-vector of a particular meek one which can be computed explicitly. Finally, we discuss the n-cube as a non-trivial example of a wild polytope with special simplex and prove that its basis polytope is the zonotope given by the Minkowski sum of the (n-1)-cube and the vector (1,...,1). Polytopes with special simplex have applications on Ehrhart theory, toric rings and were just used by Francisco Santos to construct a counter-example disproving the Hirsch conjecture.

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