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Decomposing the Secondary Cayley Polytope

T. Michiels, R. Cools
2000 Discrete & Computational Geometry  
doi:10.1007/pl00009506 fatcat:p3wrdjwmvrc3fb6ma2maetzvm4

Page 8750 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
{For the entire collection see MR 2000m:05004. } 2000m:52020 52B11 65D18 Michiels, T. (B-KUL-C; Leuven (Heverlee)) ; Cools, R. (B-KUL-C; Leuven (Heverlee)) Decomposing the secondary Cayley polytope.  ...  By embedding in a higher-dimensional space, the secondary polytope of a tuple of point configuration is shown to be Minkowski sum of (conve- niently defined) mixed secondary polytopes of the elements of  ... 

New examples of compact manifolds with holonomy Spin(7)

Robert Clancy
2011 Annals of Global Analysis and Geometry  
The essential ingredient in Joyce's construction is a Calabi-Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes the singularities.  ...  We search the class of well-formed quasismooth hypersurfaces in weighted projective spaces for suitable Calabi-Yau 4-orbifolds.  ...  For ∆ * to admit a t-invariant triangulation, we require that t must fix a face of the secondary polytope, or equivalently the fixed point set of t intersects the interior of a face of the secondary polytope  ... 
doi:10.1007/s10455-011-9254-4 fatcat:owphptgpwvcahkjjl2ad2ib7au

The Cayley trick and triangulations of products of simplices [article]

Francisco Santos
2004 arXiv   pre-print
We use the Cayley Trick to study polyhedral subdivisions of the product of two simplices.  ...  We include "Cayley Trick pictures" of all the triangulations of Δ^2×Δ^2 and Δ^2×Δ^3, as well as one non-regular triangulation of Δ^2×Δ^5 and one of Δ^3×Δ^3.  ...  Observe that 2k − 2 is the dimension of the corresponding secondary polytope, hence it is a lower bound for the number of flips of every regular triangulation.  ... 
arXiv:math/0312069v3 fatcat:slynz625tfca3a46xfro6e3bni

Proof of a Conjecture of Batyrev and Nill [article]

David Favero, Tyler L. Kelly
2016 arXiv   pre-print
We prove equivalences of derived categories for the various mirrors in the Batyrev-Borisov construction. In particular, we obtain a positive answer to a conjecture of Batyrev and Nill.  ...  The proof involves passing to an associated category of singularities and toric variation of geometric invariant theory quotients.  ...  Acknowledgments: We heartily thank Colin Diemer for suggesting that VGIT may relate to the double mirror picture and give special thanks to Charles Doran for input on this project from start to finish.  ... 
arXiv:1412.1354v5 fatcat:sy7qwjxklnb25g5megdegnskti

Proof of a conjecture of Batyrev and Nill

David Favero, Tyler L. Kelly
2017 American Journal of Mathematics  
We prove equivalences of derived categories for the various mirrors in the Batyrev-Borisov construction. In particular, we obtain a positive answer to a conjecture of Batyrev and Nill.  ...  The proof involves passing to an associated category of singularities and toric variation of geometric invariant theory quotients.  ...  Acknowledgments: We heartily thank Colin Diemer for suggesting that VGIT may relate to the double mirror picture and give special thanks to Charles Doran for input on this project from start to finish.  ... 
doi:10.1353/ajm.2017.0038 fatcat:72a4fkaqh5ea7aflojm44tfuxu

Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics [article]

Sean Timothy Paul
2010 arXiv   pre-print
As a corollary it is shown that the Mabuchi energy is bounded from below for all degenerations in G if and only if the hyperdiscriminant polytope dominates the Chow polytope for all maximal algebraic tori  ...  It is shown that the Mabuchi energy of X restricted to the Bergman metrics is completely determined by the X-hyperdiscriminant of format (n-1) and the Chow form of X.  ...  during a visit to Madison, emphasized that I should seek out the relevant polytopes.  ... 
arXiv:0811.2548v3 fatcat:lzjpnxs6azdjzivvnzfdedysu4

Bernstein's second theorem and Viro's method for sparse polynomial systems in chemistry

Karin Gatermann, Matthias Wolfrum
2005 Advances in Applied Mathematics  
Examples from applications illustrate the theoretical results.  ...  A convex polyhedral cone serves as a representative of all positive solutions of the family. We study the boundary of this cone with Bernstein's second theorem and Viro's method.  ...  Karin Gatermann was supported by the DFG by a Heisenberg scholarship.  ... 
doi:10.1016/j.aam.2004.04.003 fatcat:obmolz44pfc7pl7sv5ebybabke

Tropical discriminants

Alicia Dickenstein, Eva Maria Feichtner, Bernd Sturmfels
2007 Journal of The American Mathematical Society  
Acknowledgements We thank the Forschungsinstitut für Mathematik at ETH Zürich for hosting Alicia Dickenstein and Bernd Sturmfels in the summer of 2005.  ...  Hence, the Newton polytope of ∆ A is a Minkowski summand of the secondary polytope of A, which in turn implies that the secondary fan Σ(A) is a refinement of the normal fan of the Newton polytope of ∆  ...  The magnification on the left in Figure 3 . 3 The tropical discriminant is a subfan of the secondary fan from the secondary fan refines the fan structure dual to the Newton polytope of ∆ A .  ... 
doi:10.1090/s0894-0347-07-00562-0 fatcat:klum3kv435dlpf2wxsrbh24dua

Tropical Discriminants [article]

Alicia Dickenstein, Eva Maria Feichtner, Bernd Sturmfels
2007 arXiv   pre-print
The tropical A-discriminant, which is the tropicalization of the dual variety of the projective toric variety given by an integer matrix A, is shown to coincide with the Minkowski sum of the row space  ...  of A and of the tropicalization of the kernel of A.  ...  Acknowledgement: We thank the Forschungsinstitut für Mathematik at ETH Zürich for hosting Alicia Dickenstein and Bernd Sturmfels in the summer of 2005.  ... 
arXiv:math/0510126v3 fatcat:si7pcfxsq5e2ridygmwva2ensy

A Polyhedral Homotopy Algorithm For Real Zeros [article]

Alperen A. Ergür, Timo de Wolff
2022 arXiv   pre-print
In more technical terms; we design an algorithm that correctly counts and finds the real zeros of polynomial systems that are located in the unbounded components of the complement of the underlying A-discriminant  ...  The algorithm is targeted for polynomial systems with coefficients satisfying certain concavity conditions. It operates entirely over the real numbers and tracks the optimal number of solution paths.  ...  Polyhedral subdivisions, secondary polytope and Cayley configuration.  ... 
arXiv:1910.01957v5 fatcat:waau4s4afne7ti7hq4dxwuvuu4

Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics

Sean Paul
2012 Annals of Mathematics  
As a corollary it is shown that the Mabuchi energy is bounded from below for all degenerations in G if and only if the hyperdiscriminant polytope dominates the Chow polytope for all maximal algebraic tori  ...  It is shown that the Mabuchi energy of (X, ωF S |X ) restricted to the Bergman metrics is completely determined by the X-hyperdiscriminant of format (n − 1) and the Chow form of X.  ...  The ideas of Gang Tian have been indispensible; the whole architecture of my program was inspired by his work on special degenerations and the generalized Futaki invariant.  ... 
doi:10.4007/annals.2012.175.1.7 fatcat:fiirn5nucjcqfcph7pfla2lzfq

Book Review: On quaternions and octonions: Their geometry, arithmetic, and symmetry

John C. Baez
2005 Bulletin of the American Mathematical Society  
The 8 with integer coordinates form the vertices of a cross-polytope (the 4d analogue of an octahedron): All but one of the regular polytopes in 4 dimensions are analogues of Platonic solids in 3 dimensions  ...  The place to start is Conway and Smith. 2000 Mathematics Subject Classification. Primary 11H31, 17-01, 17A35; Secondary 17C40, 22E46, 52B11.  ... 
doi:10.1090/s0273-0979-05-01043-8 fatcat:njayhz25ofb7lmm2fvw57hqna4

Existence of unimodular triangulations - positive results [article]

Christian Haase, Andreas Paffenholz, Lindsay C. Piechnik, Francisco Santos
2017 arXiv   pre-print
We include, in particular, the first effective proof of the classical result by Knudsen-Mumford-Waterman stating that every lattice polytope has a dilation that admits a unimodular triangulation.  ...  Our proof yields an explicit (although doubly exponential) bound for the dilation factor.  ...  The Cayley difference of two lattice polytopes P 1 , P 2 ⊂ R d is the lattice polytope conv(P 1 × {0} ∪ (−P 2 ) × {1}) (compare with the definitions of Cayley sum in Sections 2.3.3 and 4.3).  ... 
arXiv:1405.1687v3 fatcat:izvgachey5aahayjct6lqubvmi

Restriction of A-Discriminants and Dual Defect Toric Varieties [article]

Raymond Curran, Eduardo Cattani
2006 arXiv   pre-print
We study the A-discriminant of toric varieties. We reduce its computation to the case of irreducible configurations and describe its behavior under specialization of some of the variables to zero.  ...  For codimension less than or equal to four, this condition is also necessary and we expect this to be the case in general.  ...  ., 1994, Chapter 10, Theorem 1.4 a) that the secondary fan Σ(A) is the normal fan to the Newton polytope N (E A ) of the principal A−determinant (we refer to (Gel ′ fand et al., 1994, Chapter 10) for the  ... 
arXiv:math/0510615v2 fatcat:jyyb22255jfjrga66rz22c6vna
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