172 research outputs found
The universality theorem for neighborly polytopes
In this note, we prove that every open primary basic semialgebraic set is
stably equivalent to the realization space of an even-dimensional neighborly
polytope. This in particular provides the final step for Mn\"ev's proof of the
universality theorem for simplicial polytopes.Comment: 5 pages, 1 figure. Small change
Realization spaces of 4-polytopes are universal
Let be a -dimensional polytope. The {\em realization space}
of~ is the space of all polytopes that are combinatorially
equivalent to~, modulo affine transformations. We report on work by the
first author, which shows that realization spaces of \mbox{4-dimensional}
polytopes can be ``arbitrarily bad'': namely, for every primary semialgebraic
set~ defined over~, there is a -polytope whose realization
space is ``stably equivalent'' to~. This implies that the realization space
of a -polytope can have the homotopy type of an arbitrary finite simplicial
complex, and that all algebraic numbers are needed to realize all -
polytopes. The proof is constructive. These results sharply contrast the
-dimensional case, where realization spaces are contractible and all
polytopes are realizable with integral coordinates (Steinitz's Theorem). No
similar universality result was previously known in any fixed dimension.Comment: 10 page
Dyck path triangulations and extendability
We introduce the Dyck path triangulation of the cartesian product of two
simplices . The maximal simplices of this
triangulation are given by Dyck paths, and its construction naturally
generalizes to produce triangulations of
using rational Dyck paths. Our study of the Dyck path triangulation is
motivated by extendability problems of partial triangulations of products of
two simplices. We show that whenever , any triangulation of
extends to a unique triangulation of
. Moreover, with an explicit construction, we
prove that the bound is optimal. We also exhibit interesting
interpretations of our results in the language of tropical oriented matroids,
which are analogous to classical results in oriented matroid theory.Comment: 15 pages, 14 figures. Comments very welcome
On a Mutation Problem for Oriented Matroids
AbstractFor uniform oriented matroids M with n elements, there is in the realizable case a sharp lower bound Lr(n) for the number mut(M) of mutations of M: Lr(n) =n≤mut(M), see Shannon [17]. Finding a sharp lower bound L(n) ≤mut(M) in the non-realizable case is an open problem for rank d≥ 4. Las Vergnas [9] conjectured that 1 ≤L(n). We study in this article the rank 4 case. Richter-Gebert [11] showed thatL (4 k) ≤ 3 k+ 1 for k≥ 2. We confirm Las Vergnas’ conjecture for n< 13. We show that L(7k+c) ≤ 5 k+c for all integersk≥ 0 and c≥ 4, and we provide a 17 element example with a mutation free element
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