50 research outputs found

    Lower Bounds for Real Solutions to Sparse Polynomial Systems

    Get PDF
    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

    Full text link
    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 Γ\Gamma with additional cohomological properties. For Γ=Z2\Gamma=\mathbb{Z}_2 we recover the conjugation spaces of Hausmann, Holm and Puppe. For Γ=U(1)\Gamma=\mathrm{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.Comment: 30 page

    A wall crossing formula for degrees of real central projections

    Full text link
    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 Z\Z-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

    Get PDF
    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

    Halving spaces and lower bounds in real enumerative geometry

    Get PDF
    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 Γ with additional cohomological properties. For Γ=Z2 we recover the conjugation spaces of Hausmann, Holm and Puppe. For Γ=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

    Geometrically bounding 3-manifold, volume and Betti number

    Full text link
    It is well known that an arbitrary closed orientable 33-manifold can be realized as the unique boundary of a compact orientable 44-manifold, that is, any closed orientable 33-manifold is cobordant to zero. In this paper, we consider the geometric cobordism problem: a hyperbolic 33-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 v≈4.3062
v \approx 4.3062\ldots be the volume of the regular right-angled hyperbolic dodecahedron in H3\mathbb{H}^{3}, for each n∈Z+n \in \mathbb{Z}_{+} and each odd integer kk in [1,5n+3][1,5n+3], we construct a closed hyperbolic 3-manifold MM with ÎČ1(M)=k\beta^1(M)=k and vol(M)=16nvvol(M)=16nv 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 description
    corecore