95 research outputs found
On the frontiers of polynomial computations in tropical geometry
We study some basic algorithmic problems concerning the intersection of
tropical hypersurfaces in general dimension: deciding whether this intersection
is nonempty, whether it is a tropical variety, and whether it is connected, as
well as counting the number of connected components. We characterize the
borderline between tractable and hard computations by proving
-hardness and #-hardness results under various
strong restrictions of the input data, as well as providing polynomial time
algorithms for various other restrictions.Comment: 17 pages, 5 figures. To appear in Journal of Symbolic Computatio
Real k-flats tangent to quadrics in R^n
Let d_{k,n} and #_{k,n} denote the dimension and the degree of the
Grassmannian G_{k,n} of k-planes in projective n-space, respectively. For each
k between 1 and n-2 there are 2^{d_{k,n}} \cdot #_{k,n} (a priori complex)
k-planes in P^n tangent to d_{k,n} general quadratic hypersurfaces in P^n. We
show that this class of enumerative problem is fully real, i.e., for each k
between 1 and n-2 there exists a configuration of d_{k,n} real quadrics in
(affine) real space R^n so that all the mutually tangent k-flats are real.Comment: 10 pages, 3 figures. Minor revisions, to appear in Proc. AM
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.Comment: Thorough revision of the first version. (13 pages, 4 figures
Radii minimal projections of polytopes and constrained optimization of symmetric polynomials
We provide a characterization of the radii minimal projections of polytopes
onto -dimensional subspaces in Euclidean space \E^n. Applied on simplices
this characterization allows to reduce the computation of an outer radius to a
computation in the circumscribing case or to the computation of an outer radius
of a lower-dimensional simplex. In the second part of the paper, we use this
characterization to determine the sequence of outer -radii of regular
simplices (which are the radii of smallest enclosing cylinders). This settles a
question which arose from the incidence that a paper by Wei{\ss}bach (1983) on
this determination was erroneous. In the proof, we first reduce the problem to
a constrained optimization problem of symmetric polynomials and then to an
optimization problem in a fixed number of variables with additional integer
constraints.Comment: Minor revisions. To appear in Advances in Geometr
- …