891 research outputs found
On k-Convex Polygons
We introduce a notion of -convexity and explore polygons in the plane that
have this property. Polygons which are \mbox{-convex} can be triangulated
with fast yet simple algorithms. However, recognizing them in general is a
3SUM-hard problem. We give a characterization of \mbox{-convex} polygons, a
particularly interesting class, and show how to recognize them in \mbox{} time. A description of their shape is given as well, which leads to
Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex
sets. Finally, we introduce the concept of generalized geometric permutations,
and show that their number can be exponential in the number of
\mbox{-convex} objects considered.Comment: 23 pages, 19 figure
How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties
Let denote the discriminant variety of degree
polynomials in one variable with at least one of its roots being of
multiplicity . We prove that the tangent cones to
span thus, revealing an extreme ruled nature of these
varieties. The combinatorics of the web of affine tangent spaces to in is directly linked to the root multiplicities
of the relevant polynomials. In fact, solving a polynomial equation
turns out to be equivalent to finding hyperplanes through a given point
P(z)\in \mathcal D_{d,1} \approx \A^d which are tangent to the discriminant
hypersurface . We also connect the geometry of the Vi\`{e}te
map \mathcal V_d: \A^d_{root} \to \A^d_{coef}, given by the elementary
symmetric polynomials, with the tangents to the discriminant varieties
.
Various -partitions provide a refinement of the stratification of \A^d_{coef} by the 's. Our main result, Theorem 7.1, describes an intricate relation
between the divisibility of polynomials in one variable and the families of
spaces tangent to various strata .Comment: 43 pages, 12 figure
Constructing matrix geometric means
In this paper, we analyze the process of "assembling" new matrix geometric
means from existing ones, through function composition or limit processes. We
show that for n=4 a new matrix mean exists which is simpler to compute than the
existing ones. Moreover, we show that for n>4 the existing proving strategies
cannot provide a mean computationally simpler than the existing ones
Empty Rectangles and Graph Dimension
We consider rectangle graphs whose edges are defined by pairs of points in
diagonally opposite corners of empty axis-aligned rectangles. The maximum
number of edges of such a graph on points is shown to be 1/4 n^2 +n -2.
This number also has other interpretations:
* It is the maximum number of edges of a graph of dimension
\bbetween{3}{4}, i.e., of a graph with a realizer of the form
\pi_1,\pi_2,\ol{\pi_1},\ol{\pi_2}.
* It is the number of 1-faces in a special Scarf complex.
The last of these interpretations allows to deduce the maximum number of
empty axis-aligned rectangles spanned by 4-element subsets of a set of
points. Moreover, it follows that the extremal point sets for the two problems
coincide.
We investigate the maximum number of of edges of a graph of dimension
, i.e., of a graph with a realizer of the form
\pi_1,\pi_2,\pi_3,\ol{\pi_3}. This maximum is shown to be .
Box graphs are defined as the 3-dimensional analog of rectangle graphs. The
maximum number of edges of such a graph on points is shown to be
- …