50 research outputs found
Lower Bounds for Real Solutions to Sparse Polynomial Systems
We show how to construct sparse polynomial systems that have non-trivial
lower bounds on their numbers of real solutions. These are unmixed systems
associated to certain polytopes. For the order polytope of a poset P this lower
bound is the sign-imbalance of P and it holds if all maximal chains of P have
length of the same parity. This theory also gives lower bounds in the real
Schubert calculus through sagbi degeneration of the Grassmannian to a toric
variety, and thus recovers a result of Eremenko and Gabrielov.Comment: 31 pages. Minor revision
Halving spaces and lower bounds in real enumerative geometry
We develop the theory of halving spaces to obtain lower bounds in real
enumerative geometry. Halving spaces are topological spaces with an action of a
Lie group with additional cohomological properties. For
we recover the conjugation spaces of Hausmann, Holm and
Puppe. For we obtain the circle spaces. We show that
real even and quaternionic partial flag manifolds are circle spaces leading to
non-trivial lower bounds for even real and quaternionic Schubert problems. To
prove that a given space is a halving space, we generalize results of Borel and
Haefliger on the cohomology classes of real subvarieties and their
complexifications. The novelty is that we are able to obtain results in
rational cohomology instead of modulo 2. The equivariant extension of the
theory of circle spaces leads to generalizations of the results of Borel and
Haefliger on Thom polynomials.Comment: 30 page
A wall crossing formula for degrees of real central projections
The main result is a wall crossing formula for central projections defined on
submanifolds of a real projective space. Our formula gives the jump of the
degree of such a projection when the center of the projection varies. The fact
that the degree depends on the projection is a new phenomenon, specific to real
algebraic geometry. We illustrate this phenomenon in many interesting
situations. The crucial assumption on the class of maps we consider is relative
orientability, a condition which allows us to define a -valued degree map
in a coherent way. We end the article with several examples, e.g. the pole
placement map associated with a quotient, the Wronski map, and a new version of
the real subspace problem.Comment: 29 pages. First revised version: The proof of the "wall-crossing
formula" is now more conceptional. We prove new general properties of the set
of values of the degree map on the set of central projections. Second revised
version: minor corrections. To appear in International Journal of Mathematic
Products of Foldable Triangulations
Regular triangulations of products of lattice polytopes are constructed with
the additional property that the dual graphs of the triangulations are
bipartite. The (weighted) size difference of this bipartition is a lower bound
for the number of real roots of certain sparse polynomial systems by recent
results of Soprunova and Sottile [Adv. Math. 204(1):116-151, 2006]. Special
attention is paid to the cube case.Comment: new title; several paragraphs reformulated; 23 page
Recommended from our members
Tropical Geometry: new directions
The workshop "Tropical Geometry: New Directions" was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject, notably, to new phenomena that
have opened themselves in the course of the last 4 years. This includes, in particular, refined enumerative
geometry (using positive integer q-numbers instead of positive integer numbers), unexpected appearance of tropical curves in scaling limits of Abelian sandpile models, as well as a significant progress
in more traditional areas of tropical research, such as tropical
moduli spaces, tropical homology and tropical correspondence theorems
Halving spaces and lower bounds in real enumerative geometry
We develop the theory of halving spaces to obtain lower bounds in real enumerative geometry. Halving spaces are topological spaces with an action of a Lie group Î with additional cohomological properties. For Î=Z2 we recover the conjugation spaces of Hausmann, Holm and Puppe. For Î=U(1) we obtain the circle spaces. We show that real even and quaternionic partial flag manifolds are circle spaces leading to non-trivial lower bounds for even real and quaternionic Schubert problems. To prove that a given space is a halving space, we generalize results of Borel and Haefliger on the cohomology classes of real subvarieties and their complexifications. The novelty is that we are able to obtain results in rational cohomology instead of modulo 2. The equivariant extension of the theory of circle spaces leads to generalizations of the results of Borel and Haefliger on Thom polynomials
Recommended from our members
Tropical Aspects in Geometry, Topology and Physics
The workshop Tropical Aspects in Geometry, Topology and Physics was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject. The development of tropical geometry is based on deep links between problems in real and complex enumerative geometry, symplectic geometry, quantum fields theory, mirror symmetry, dynamical systems and other research areas. On the other hand, new interesting phenomena discovered in the framework of tropical geometry (like refined tropical enumerative invariants) pose the problem of a conceptual understanding of these phenomena in the âclassicalâ geometry and mathematical physics
Geometrically bounding 3-manifold, volume and Betti number
It is well known that an arbitrary closed orientable -manifold can be
realized as the unique boundary of a compact orientable -manifold, that is,
any closed orientable -manifold is cobordant to zero. In this paper, we
consider the geometric cobordism problem: a hyperbolic -manifold is
geometrically bounding if it is the only boundary of a totally geodesic
hyperbolic 4-manifold. However, there are very rare geometrically bounding
closed hyperbolic 3-manifolds according to the previous research [11,13]. Let
be the volume of the regular right-angled hyperbolic
dodecahedron in , for each and each odd
integer in , we construct a closed hyperbolic 3-manifold with
and that bounds a totally geodesic hyperbolic
4-manifold. The proof uses small cover theory over a sequence of linearly-glued
dodecahedra and some results of Kolpakov-Martelli-Tschantz [9].Comment: the latest version that adjust some figures and add more detail
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