276 research outputs found
The Hirsch conjecture holds for normal flag complexes
Using an intuition from metric geometry, we prove that any flag and normal
simplicial complex satisfies the non-revisiting path conjecture. As a
consequence, the diameter of its facet-ridge graph is smaller than the number
of vertices minus the dimension, as in the Hirsch conjecture. This proves the
Hirsch conjecture for all flag polytopes, and more generally, for all
(connected) flag homology manifolds.Comment: 9 pages, 1 figure; to appear in Mathematics of Operations Researc
On the dual graph of Cohen-Macaulay algebras
Given a projective algebraic set X, its dual graph G(X) is the graph whose
vertices are the irreducible components of X and whose edges connect components
that intersect in codimension one. Hartshorne's connectedness theorem says that
if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We
present two quantitative variants of Hartshorne's result:
1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where
r is the Castelnuovo-Mumford regularity of X. (The bound is best possible; for
coordinate arrangements, it yields an algebraic extension of Balinski's theorem
for simplicial polytopes.)
2) If X is a canonically embedded arrangement of lines no three of which meet
in the same point, then the diameter of the graph G(X) is not larger than the
codimension of X. (The bound is sharp; for coordinate arrangements, it yields
an algebraic expansion on the recent combinatorial result that the Hirsch
conjecture holds for flag normal simplicial complexes.)Comment: Minor changes throughout, Remark 4.1 expanded, to appear in IMR
Recent progress on the combinatorial diameter of polytopes and simplicial complexes
The Hirsch conjecture, posed in 1957, stated that the graph of a
-dimensional polytope or polyhedron with facets cannot have diameter
greater than . The conjecture itself has been disproved, but what we
know about the underlying question is quite scarce. Most notably, no polynomial
upper bound is known for the diameters that were conjectured to be linear. In
contrast, no polyhedron violating the conjecture by more than 25% is known.
This paper reviews several recent attempts and progress on the question. Some
work in the world of polyhedra or (more often) bounded polytopes, but some try
to shed light on the question by generalizing it to simplicial complexes. In
particular, we include here our recent and previously unpublished proof that
the maximum diameter of arbitrary simplicial complexes is in and
we summarize the main ideas in the polymath 3 project, a web-based collective
effort trying to prove an upper bound of type nd for the diameters of polyhedra
and of more general objects (including, e. g., simplicial manifolds).Comment: 34 pages. This paper supersedes one cited as "On the maximum diameter
of simplicial complexes and abstractions of them, in preparation
Stellar theory for flag complexes
Refining a basic result of Alexander, we show that two flag simplicial
complexes are piecewise linearly homeomorphic if and only if they can be
connected by a sequence of flag complexes, each obtained from the previous one
by either an edge subdivision or its inverse. For flag spheres we pose new
conjectures on their combinatorial structure forced by their face numbers,
analogous to the extremal examples in the upper and lower bound theorems for
simplicial spheres. Furthermore, we show that our algorithm to test the
conjectures searches through the entire space of flag PL spheres of any given
dimension.Comment: 12 pages, 2 figures. Notation unified and presentation of proofs
improve
A counterexample to the Hirsch conjecture
The Hirsch Conjecture (1957) stated that the graph of a -dimensional
polytope with facets cannot have (combinatorial) diameter greater than
. That is, that any two vertices of the polytope can be connected by a
path of at most edges.
This paper presents the first counterexample to the conjecture. Our polytope
has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope
with 48 facets which violates a certain generalization of the -step
conjecture of Klee and Walkup.Comment: 28 pages, 10 Figures: Changes from v2: Minor edits suggested by
referees. This version has been accepted in the Annals of Mathematic
On the diameter of an ideal
We begin the study of the notion of diameter of an ideal I of a polynomial
ring S over a field, an invariant measuring the distance between the minimal
primes of I. We provide large classes of Hirsch ideals, i.e. ideals with
diameter not larger than the codimension, such as: quadratic radical ideals of
codimension at most 4 and such that S/I is Gorenstein, or ideals admitting a
square-free complete intersection initial ideal
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