Newton polyhedra and zeros of systems of exponential sums

Kazarnovskii, Boris

doi:10.1007/bf01083691

Cited by 24 publications

(21 citation statements)

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“…It is so called Kazarnovskii's pseudovolume. It was introduced and studied by B. Kazarnovskii [30], [31] in order to write down a formula for the number of zeros of a system of exponential sums in terms of their Newton polytopes. His results generalize in some sense the well known results of D. Bernstein [7] and A. Kouchnirenko [35] on the number of zeros of a system of polynomial equations (see also [18]).…”

Section: B) a Valuation φ Is Called Continuous If It Is Continuous Wimentioning

confidence: 99%

“…The proof of this result is based on the classification of unitarily invariant valuations (Theorem 2.1.1). Now let us recall the definition of Kazarnovskii's pseudovolume following [30], [31]. Let C n be Hermitian space with the Hermitian scalar product (·, ·).…”

Section: Kazarnovskii's Pseudovolumementioning

confidence: 99%

“…The first part of the proposition (the continuity) is a standard fact from the theory of plurisubharmonic functions originally due to ChernLevine-Nirenberg [14] (see also [30], [31]). The unitary invariance, translation invariance, and the homogeneity of degree n are obvious.…”

Section: Kazarnovskii's Pseudovolumementioning

confidence: 99%

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Hard Lefschetz Theorem for Valuations, Complex Integral Geometry, and Unitarily Invariant Valuations

Alesker¹

2003

J. Differential Geom.

112

183

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We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new integral geometric formulas for real submanifolds in Hermitian spaces generalizing the classical kinematic formulas in Euclidean spaces due to Poincaré, Chern, Santaló, and others. 0 Introduction.In this paper we obtain new results on the structure of the space of even translation invariant continuous valuations on convex sets. In particular we prove a version of hard Lefschetz theorem for them and introduce certain natural duality operator which establishes an isomorphism between the space of such valuations on a linear space V and on its dual V * (with an appropriate twisting). Then we obtain an explicit geometric classification of unitarily invariant translation invariant continuous valuations on a Hermitian space C n . This classification is used to deduce new integral geometric formulas for real submanifolds in Hermitian spaces generalizing the classical kinematic formulas in Euclidean spaces due to Poincaré, Chern, Santaló, and others.

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Section: B) a Valuation φ Is Called Continuous If It Is Continuous Wimentioning

confidence: 99%

Section: Kazarnovskii's Pseudovolumementioning

confidence: 99%

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Hard Lefschetz Theorem for Valuations, Complex Integral Geometry, and Unitarily Invariant Valuations

Alesker¹

2003

J. Differential Geom.

112

183

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“…In Section 4.1 we remind Kazarnovskii's pseudovolume and its generalizations following [26,27]. In Section 4.2 we obtain quaternionic analogues of these valuations, i.e.…”

Section: Introductionmentioning

confidence: 99%

Valuations on convex sets, non-commutative determinants, and pluripotential theory

Alesker

2005

Advances in Mathematics

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A new method of constructing translation invariant continuous valuations on convex subsets of the quaternionic space H n is presented. In particular new examples of Sp(n)Sp(1)-invariant translation invariant continuous valuations are constructed. This method is based on the theory of plurisubharmonic functions of quaternionic variables developed by the author in two previous papers (Bull. The goal of this paper is to present a new method of constructing translation invariant continuous valuations on convex subsets of the quaternionic space H n . As an application of this method we obtain new examples of Sp(n)Sp(1)-invariant translation invariant continuous valuations. This method is based on the theory of plurisubharmonic functions of quaternionic variables developed by the author in two previous papers [5,6]. The main results of the paper are Theorem 4.2.1 and its immediate Corollary 4.2.2. Let us remind basic notions of the theory of valuations on convex sets referring for more details to the surveys by McMullen [30] and McMullen and Schneider [31]. Let V be a finite-dimensional real vector space. Let K(V ) denote the class of all convex compact subsets of V.

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“…It should be noted that a complex (and in fact the original) version of the pseudo-volume using the complex Hessian was first considered in the context of convexity (though not of valuations) by Kazarnovskiȋ [35], [36]. The quaternionic version of the pseudo-volume using the quaternionic Hessian was constructed by the author in [11].…”

Section: Examplementioning

confidence: 99%

Plurisubharmonic Functions on the Octonionic Plane and Spin(9)-Invariant Valuations on Convex Sets

Alesker

2008

J Geom Anal

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Newton polyhedra and zeros of systems of exponential sums

Cited by 24 publications

References 1 publication

Hard Lefschetz Theorem for Valuations, Complex Integral Geometry, and Unitarily Invariant Valuations

Hard Lefschetz Theorem for Valuations, Complex Integral Geometry, and Unitarily Invariant Valuations

Valuations on convex sets, non-commutative determinants, and pluripotential theory

Plurisubharmonic Functions on the Octonionic Plane and Spin(9)-Invariant Valuations on Convex Sets

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