11 research outputs found
A note on order-type homogeneous point sets
Let OT_d(n) be the smallest integer N such that every N-element point
sequence in R^d in general position contains an order-type homogeneous subset
of size n, where a set is order-type homogeneous if all (d+1)-tuples from this
set have the same orientation. It is known that a point sequence in R^d that is
order-type homogeneous forms the vertex set of a convex polytope that is
combinatorially equivalent to a cyclic polytope in R^d. Two famous theorems of
Erdos and Szekeres from 1935 imply that OT_1(n) = Theta(n^2) and OT_2(n) =
2^(Theta(n)). For d \geq 3, we give new bounds for OT_d(n). In particular:
1. We show that OT_3(n) = 2^(2^(Theta(n))), answering a question of
Eli\'a\v{s} and Matou\v{s}ek.
2. For d \geq 4, we show that OT_d(n) is bounded above by an exponential
tower of height d with O(n) in the topmost exponent
Interpolation of toric varieties
Let be a -dimensional variety in -dimensional
projective space. Let be a positive integer such that .
Consider the following interpolation problem: does there exist a variety
of dimension , with ,
such that the tangent space to at a point is equal to the th
osculating space to at , for almost all points ? In this paper
we consider this question in the toric setting. We prove that if is toric,
then there is a unique toric variety solving the above interpolation
problem. We identify in the general case and we explicitly compute some of
its invariants when is a toric curve.Comment: 12 pages, 1 figur
Constructing neighborly polytopes and oriented matroids
A d-polytope P is neighborly if every subset of b d 2 c vertices is a face of P. In 1982, Shemer introduced a sewing construction that allows to add a vertex to a neighborly polytope in such a way as to obtain a new neighborly polytope. With this, he constructed superexponentially many different neighborly polytopes. The concept
of neighborliness extends naturally to oriented matroids. Duals of neighborly oriented matroids also have a nice
characterization: balanced oriented matroids. In this paper, we generalize Shemer’s sewing construction to oriented
matroids, providing a simpler proof. Moreover we provide a new technique that allows to construct balanced oriented
matroids. In the dual setting, it constructs a neighborly oriented matroid whose contraction at a particular vertex is a prescribed neighborly oriented matroid. We compare the families of polytopes that can be constructed with both methods, and show that the new construction allows to construct many new polytopes.Peer ReviewedPostprint (published version
Sign variation, the Grassmannian, and total positivity
The totally nonnegative Grassmannian is the set of k-dimensional subspaces V
of R^n whose nonzero Pluecker coordinates all have the same sign. Gantmakher
and Krein (1950) and Schoenberg and Whitney (1951) independently showed that V
is totally nonnegative iff every vector in V, when viewed as a sequence of n
numbers and ignoring any zeros, changes sign at most k-1 times. We generalize
this result from the totally nonnegative Grassmannian to the entire
Grassmannian, showing that if V is generic (i.e. has no zero Pluecker
coordinates), then the vectors in V change sign at most m times iff certain
sequences of Pluecker coordinates of V change sign at most m-k+1 times. We also
give an algorithm which, given a non-generic V whose vectors change sign at
most m times, perturbs V into a generic subspace whose vectors also change sign
at most m times. We deduce that among all V whose vectors change sign at most m
times, the generic subspaces are dense. These results generalize to oriented
matroids. As an application of our results, we characterize when a generalized
amplituhedron construction, in the sense of Arkani-Hamed and Trnka (2013), is
well defined. We also give two ways of obtaining the positroid cell of each V
in the totally nonnegative Grassmannian from the sign patterns of vectors in V.Comment: 28 pages. v2: We characterize when a generalized amplituhedron
construction is well defined, in new Section 4 (the previous Section 4 is now
Section 5); v3: Final version to appear in J. Combin. Theory Ser.
Constructing neighborly polytopes and oriented matroids
A -polytope is neighborly if every subset of vertices is a face of . In 1982, Shemer introduced a sewing construction that allows to add a vertex to a neighborly polytope in such a way as to obtain a new neighborly polytope. With this, he constructed superexponentially many different neighborly polytopes. The concept of neighborliness extends naturally to oriented matroids. Duals of neighborly oriented matroids also have a nice characterization: balanced oriented matroids. In this paper, we generalize Shemer's sewing construction to oriented matroids, providing a simpler proof. Moreover we provide a new technique that allows to construct balanced oriented matroids. In the dual setting, it constructs a neighborly oriented matroid whose contraction at a particular vertex is a prescribed neighborly oriented matroid. We compare the families of polytopes that can be constructed with both methods, and show that the new construction allows to construct many new polytopes
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Geometric, Algebraic, and Topological Combinatorics
The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics"
was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle),
Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered
a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics
with geometric flavor, and Topological Combinatorics. Some of the
highlights of the conference included (1) Karim Adiprasito presented his
very recent proof of the -conjecture for spheres (as a talk and as a "Q\&A"
evening session) (2) Federico Ardila gave an overview on "The geometry of matroids",
including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz
Cyclic polytopes and oriented matroids
Consider the moment curve in the real Euclidean space R d defined parametrically by the map γ: R → R d, t ↦ → γ(t) =(t, t 2,...,t d). The cyclic d-polytope Cd(t1,...,tn) is the convex hull of the n, n> d, different points on this curve. The matroidal analogues are the alternating oriented uniform matroids. A polytope [resp. matroid polytope] is called cyclic if its face lattice is isomorphic to that of Cd(t1,...,tn). We give combinatorial and geometrical characterizations of cyclic [matroid] polytopes. A simple evenness criterion determining the facets of Cd(t1,...,tn) was given by David Gale. We characterize the admissible orderings of the vertices of the cyclic polytope, i.e., those linear orderings of the vertices for which Gale’s evenness criterion holds. Proofs give a systematic account on an oriented matroid approach to cyclic polytopes. ∗ 1991 Mathematics Subject Classification: Primary 05B35, 52A25. Keywords: Cyclic [matroid] polytopes, neighborly polytopes, simplicial polytopes, moment curves, d-th cyclic curves, d-th order curves, Gale evenness criterion, admissible orderings, oriented matroids, alternating [oriented] uniform matroids, inseparability graphs