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Decomposing the Secondary Cayley Polytope

2000
*
Discrete & Computational Geometry
*

##
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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 ...##
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New examples of compact manifolds with holonomy Spin(7)

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*...

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

2004
*
arXiv
*
pre-print

We use

arXiv:math/0312069v3
fatcat:slynz625tfca3a46xfro6e3bni
*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. ...##
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Proof of a Conjecture of Batyrev and Nill
[article]

2016
*
arXiv
*
pre-print

We prove equivalences of derived categories for

arXiv:1412.1354v5
fatcat:sy7qwjxklnb25g5megdegnskti
*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. ...##
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Proof of a conjecture of Batyrev and Nill

2017
*
American Journal of Mathematics
*

We prove equivalences of derived categories for

doi:10.1353/ajm.2017.0038
fatcat:72a4fkaqh5ea7aflojm44tfuxu
*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. ...##
###
Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics
[article]

2010
*
arXiv
*
pre-print

As a corollary it is shown that

arXiv:0811.2548v3
fatcat:lzjpnxs6azdjzivvnzfdedysu4
*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*. ...##
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Bernstein's second theorem and Viro's method for sparse polynomial systems in chemistry

2005
*
Advances in Applied Mathematics
*

Examples from applications illustrate

doi:10.1016/j.aam.2004.04.003
fatcat:obmolz44pfc7pl7sv5ebybabke
*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. ...##
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Tropical discriminants

2007
*
Journal of The American Mathematical Society
*

Acknowledgements We thank

doi:10.1090/s0894-0347-07-00562-0
fatcat:klum3kv435dlpf2wxsrbh24dua
*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 . ...##
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Tropical Discriminants
[article]

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. ...

##
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A Polyhedral Homotopy Algorithm For Real Zeros
[article]

2022
*
arXiv
*
pre-print

In more technical terms; we design an algorithm that correctly counts and finds

arXiv:1910.01957v5
fatcat:waau4s4afne7ti7hq4dxwuvuu4
*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. ...##
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Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics

2012
*
Annals of Mathematics
*

As a corollary it is shown that

doi:10.4007/annals.2012.175.1.7
fatcat:fiirn5nucjcqfcph7pfla2lzfq
*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. ...##
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Book Review: On quaternions and octonions: Their geometry, arithmetic, and symmetry

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. ...

##
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Existence of unimodular triangulations - positive results
[article]

2017
*
arXiv
*
pre-print

We include, in particular,

arXiv:1405.1687v3
fatcat:izvgachey5aahayjct6lqubvmi
*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). ...##
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Restriction of A-Discriminants and Dual Defect Toric Varieties
[article]

2006
*
arXiv
*
pre-print

We study

arXiv:math/0510615v2
fatcat:jyyb22255jfjrga66rz22c6vna
*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*...
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