共查询到20条相似文献,搜索用时 15 毫秒
1.
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 the sagbi degeneration of the Grassmannian to a toric variety, and thus recovers a result of Eremenko and Gabrielov. 相似文献
2.
Jan Draisma 《Journal of Pure and Applied Algebra》2008,212(2):349-363
Tropical geometry yields good lower bounds, in terms of certain combinatorial-polyhedral optimisation problems, on the dimensions of secant varieties. The approach is especially successful for toric varieties such as Segre-Veronese embeddings. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that all Veronese embeddings of the projective plane except for the quadratic one and the quartic one are non-defective; and indeed, no Segre-Veronese embeddings are known where the tropical lower bound does not give the correct dimension. Short self-contained introductions to secant varieties and the required tropical geometry are included. 相似文献
3.
Alan Stapledon 《Advances in Mathematics》2011,(4):3622
Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of numerous classical results, and give applications to the Ehrhart theory of rational polytopes and centrally symmetric polytopes. We also recover a character formula of Procesi, Dolgachev, Lunts and Stembridge for the action of a Weyl group on the cohomology of a toric variety associated to a root system. 相似文献
4.
For each dimension d, d-dimensional integral simplices with exactly one interior integral point have bounded volume. This was first shown by Hensley. Explicit volume bounds were determined by Hensley, Lagarias and Ziegler, Pikhurko, and Averkov. In this paper we determine the exact upper volume bound for such simplices and characterize the volume-maximizing simplices. We also determine the sharp upper bound on the coefficient of asymmetry of an integral polytope with a single interior integral point. This result confirms a conjecture of Hensley from 1983. Moreover, for an integral simplex with precisely one interior integral point, we give bounds on the volumes of its faces, the barycentric coordinates of the interior integral point and its number of integral points. Furthermore, we prove a bound on the lattice diameter of integral polytopes with a fixed number of interior integral points. The presented results have applications in toric geometry and in integer optimization. 相似文献
5.
Karim Adiprasito Eran Nevo José Alejandro Samper 《Geometric And Functional Analysis》2016,26(2):359-378
We resolve a conjecture of Kalai relating approximation theory of convex bodies by simplicial polytopes to the face numbers and primitive Betti numbers of these polytopes and their toric varieties. The proof uses higher notions of chordality. Further, for C 2-convex bodies, asymptotically tight lower bounds on the g-numbers of the approximating polytopes are given, in terms of their Hausdorff distance from the convex body. 相似文献
6.
We give new weighted decompositions for simple polytopes, generalizing previous results of Lawrence-Varchenko and Brianchon-Gram. We start with Witten's non-abelian localization principle in equivariant cohomology for the norm-square of the moment map in the context of toric varieties to obtain a decomposition for Delzant polytopes. Then, by a purely combinatorial argument, we show that this formula holds for any simple polytope. As an application, we study Euler-Maclaurin formulas. 相似文献
7.
Naoki Fujita 《代数通讯》2018,46(6):2666-2692
The theory of Newton-Okounkov polytopes is a generalization of that of Newton polytopes for toric varieties, and gives a systematic method of constructing toric degenerations of projective varieties. In the case of Schubert varieties, their Newton-Okounkov polytopes are deeply connected with representation theory. Indeed, Littelmann’s string polytopes and Nakashima-Zelevinsky’s polyhedral realizations are obtained as Newton-Okounkov polytopes of Schubert varieties. In this paper, we apply the folding procedure to a Newton-Okounkov polytope of a Schubert variety, which relates Newton-Okounkov polytopes of Schubert varieties of different types. As an application, we obtain a new interpretation of Kashiwara’s similarity of crystal bases. 相似文献
8.
Alexandr V. Kuzminykh 《Journal of Geometry》2004,79(1-2):134-145
A family of convex bodies in Ed is called neighborly if the
intersection of every two of them is (d-1)-dimensional. In the present paper we
prove that there is an infinite neighborly family of centrally symmetric convex bodies
in Ed, d 3, such that every two of them are affinely equivalent
(i.e., there is an affine transformation mapping one of them onto another), the
bodies have large groups of affine automorphisms, and the volumes of the bodies are
prescribed. We also prove that there is an infinite neighborly family of centrally
symmetric convex bodies in Ed such that the bodies have large groups of
symmetries. These two results are answers to a problem of B. Grünbaum (1963). We
prove also that there exist arbitrarily large neighborly families of similar convex
d-polytopes in Ed with prescribed diameters and with arbitrarily large
groups of symmetries of the polytopes. 相似文献
9.
10.
We give a lower bound for the number of vertices of a generald-dimensional polytope with a given numberm ofi-faces for eachi = 0,..., d/2 – 1. The tightness of those bounds is proved using McMullen's conditions. Form greater than a small constant, those lower bounds are attained by simpliciali-neighbourly polytopes. 相似文献
11.
Chuanming Zong 《Periodica Mathematica Hungarica》1995,30(3):233-238
This article shows an inequality concerning blocking numbers and Hadwiger's covering numbers and presents a strange phenomenon concerning kissing numbers and blocking numbers. As a simple corollary, we can improve the known upper bounds for Hadwiger's covering numbers ford-dimensional centrally symmetric convex bodies to 3
d
–1. 相似文献
12.
An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers. 相似文献
13.
Daniele Mundici 《Annals of Pure and Applied Logic》2011,162(12):1035-1047
We classify every finitely axiomatizable theory in infinite-valued propositional ?ukasiewicz logic by an abstract simplicial complex (V,Σ) equipped with a weight function ω:V→{1,2,…}. Using the W?odarczyk–Morelli solution of the weak Oda conjecture for toric varieties, we then construct a Turing computable one–one correspondence between (Alexander) equivalence classes of weighted abstract simplicial complexes, and equivalence classes of finitely axiomatizable theories, two theories being equivalent if their Lindenbaum algebras are isomorphic. We discuss the relationship between our classification and Markov’s undecidability theorem for PL-homeomorphism of rational polyhedra. 相似文献
14.
Centrally symmetric generators in toric Fano varieties 总被引:1,自引:1,他引:0
We give a structure theorem for n-dimensional smooth toric Fano varieties whose associated polytope has ``many' pairs of centrally symmetric vertices.
Mathematics Subject Classification (2000):14J45, 14M25, 52B20 相似文献
15.
Bernt Ivar Utstøl Nødland 《Journal of Pure and Applied Algebra》2018,222(3):508-533
We use Matsui and Takeuchi's formula for toric A-discriminants to give algorithms for computing local Euler obstructions and dual degrees of toric surfaces and 3-folds. In particular, we consider weighted projective spaces. As an application we give counterexamples to a conjecture by Matsui and Takeuchi. As another application we recover the well-known fact that the only defective normal toric surfaces are cones. 相似文献
16.
We give the lower bound on the number of sharp shadow-boundaries of convexd-polytopes (or unbounded convex polytopal sets) withn facets. The polytopes (sets) attaining these bounds are characterized. Additionally, our results will be transferred to the dual theory.The research work of the first author was (partially) supported by Hungarian National Foundation for Scientific Research, grant no. 1812. 相似文献
17.
Birkett Huber Jörg Rambau Francisco Santos 《Journal of the European Mathematical Society》2000,2(2):179-198
In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving
bijection between the posets of coherent mixed subdivisions of a Minkowski sum ?1+...+?
r
of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding ?(?1,...,?
r
). In this paper we extend this correspondence in a natural way to cover also non-coherent subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions of Lawrence
polytopes provides a new independent proof of the Bohne-Dress theorem on zonotopal tilings. This application uses a combinatorial
characterization of lifting subdivisions, also originally proved by Santos.
Received February 18, 1999 / final version received January 25, 2000?Published online May 22, 2000 相似文献
18.
19.
Jörg Gretenkort Peter Kleinschmidt Bernd Sturmfels 《Discrete and Computational Geometry》1990,5(1):255-262
We prove that the combinatorial types of those cone systems which correspond to complete smooth toric varieties are more restricted than for complete toric varieties: the toric varieties corresponding to essentially all types of cyclic polytopes possess singularities. This yields a negative answer to a problem stated by G. Ewald. Some consequences and problems concerning mathematical programming and the rational cohomology of smooth toric varieties are discussed.The research of P. Kleinschmidt was supported in part by the Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minnesota, USA. 相似文献
20.
In this paper, we give an algebro-geometric characterization of Cayley polytopes. As a special case, we also characterize lattice polytopes with lattice width one by using Seshadri constants. 相似文献