Lectures on Polytopes

Ziegler, Günter M.

doi:10.1007/978-1-4613-8431-1

Cited by 2,186 publications

(2,020 citation statements)

References 169 publications

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“…Recall that the Newton polytope of a polynomial is the convex hull of all exponent vectors appearing in the expansion of that polynomial, and a polytope is a zonotope if it is the image of a standard cube under a linear map; see Chapter 7 of [106] for more on zonotopes. We are here considering the zonotope …”

Section: The Input For Thatmentioning

confidence: 99%

Lectures on Algebraic Statistics

2009

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Section: The Input For Thatmentioning

confidence: 99%

Lectures on Algebraic Statistics

2009

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“…Clearly, h P (t) encodes the same information as the f -polynomial, but additionally h P (t) is a unimodal, palindromic polynomial with non-negative, integral coefficients (see e.g. [28,Sect. 8.3]).…”

Section: Rigidity With Symmetry and Flag-vector Inequalitiesmentioning

confidence: 99%

On Kalai’s Conjectures Concerning Centrally Symmetric Polytopes

Sanyal

Werner

Ziegler

2008

Discrete Comput Geom

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In 1989 Kalai stated the three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the "3 d -conjecture". It is well-known that the three conjectures hold in dimensions d ≤ 3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d ≥ 5.

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“…Let us explain fundamental notions in the theory of convex polyhedra, which are necessary for our investigation. Refer to [32] for general theory of convex polyhedra. For (a, l) ∈ R n ×R, let H(a, l) and H + (a, l) be a hyperplane and a closed halfspace in R n defined by…”

Section: Newton Polyhedra and The Classê(u)mentioning

confidence: 99%

“…The Newton polyhedron Γ + (f ) of f is defined to be the convex hull of the set {α + R n + : c α = 0}. It is known that the Newton polyhedron is a polyhedron (see [32]). The union of the compact faces of the Newton polyhedron Γ + (f ) is called the Newton diagram Γ(f ) of f , while the topological boundary of Γ + (f ) is denoted by ∂Γ + (f ).…”

Section: Newton Polyhedra and The Classê(u)mentioning

confidence: 99%

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Newton polyhedra and weighted oscillatory integrals with smooth phases

Kamimoto

Nose

2015

Trans. Amer. Math. Soc.

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Abstract. In his seminal paper, A. N. Varchenko precisely investigates the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase. He expresses the order of this term by means of the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize and improve his result. We are especially interested in the cases that the phase is smooth and that the amplitude has a zero at a critical point of the phase. In order to exactly treat the latter case, a weight function is introduced in the amplitude. Our results show that the optimal rates of decay for weighted oscillatory integrals, whose phases and weights are contained in a certain class of smooth functions including the real analytic class, can be expressed by the Newton distance and multiplicity defined in terms of geometrical relationship of the Newton polyhedra of the phase and the weight. We also compute explicit formulae of the coefficient of the leading term of the asymptotic expansion in the weighted case. Our method is based on the resolution of singularities constructed by using the theory of toric varieties, which naturally extends the resolution of Varchenko. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation. The investigation of this paper improves on the earlier joint work with K. Cho.

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Lectures on Polytopes

Abstract: ISBN 978-0-387-943 -(hardcover) ISBN 978-1-4613-8431-1 (eBook) ISBN 978----(softcover)

Cited by 2,186 publications

References 169 publications

Lectures on Algebraic Statistics

Lectures on Algebraic Statistics

On Kalai’s Conjectures Concerning Centrally Symmetric Polytopes

Newton polyhedra and weighted oscillatory integrals with smooth phases

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