Newton polyhedra and zeros of systems of exponential sums
type: Item
instance of: scholarly article
title: Newton polyhedra and zeros of systems of exponential sums
zbMATH Open document ID: 0593.32001
DOI: 10.1007/BF01083691
publication date: 1984-01-01T00:00:00Z
review text: An exponential sum (e.s.) in \({\mathbb{C}}^ n\) is defined as \[ s(z)=\sum C\lambda \exp (z,\lambda),\quad z\in {\mathbb{C}}^ n,\quad C\lambda \in {\mathbb{C}}, \] where the complex index \(\lambda\) runs through a finite set (called the spectrum of the e.s.) consisting of points of the n- dimensional complex spaces of indices dual to the complex space in which the zeros z lie. The author shows that for \(1\leq k\leq n\), the zeros of a system of k exponential sums in \({\mathbb{C}}^ n\) are asymptotically distributed along a star formed by cones of real codimension k, and he calculates the density of this distribution. The star along which the zeros of the system are distributed consists of the (2n-k)-dimensional cones dual to the faces of the resulting k-dimensional skeleton, and for \(t\to \infty\) the flow has an asymptotic expansion of the form \(\Phi_ kt^ k+ \Phi^ 1_ kt^{k-1}+ \Phi^ 2_ kt^{k-2}+...,\) where the flow \(\Phi_ k\) is equal to the value on the Newton polyhedra of a certain k-linear symmetric function of convex solids in space of indices with flows of order 2k defined in the space of zeros.
Mathematics Subject Classification ID: 32A10
Mathematics Subject Classification ID: 32A20
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 3953218
zbMATH Keywords: zeros of systems of exponential sums
zbMATH Keywords: Newton polyhedra
author: B. Ya. Kazarnovskii
reviewed by: Hussain S. Nur
MaRDI profile type: Publication
cites work: Q3326007
cites work: Q3322392
cites work: The noncommutative Markovian property
cites work: Complex differential and integral geometry and curvature integrals associated to singularities of complex analytic varieties
full work available at URL: https://doi.org/10.1007/bf01083691
OpenAlex ID: W1966155770
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 3953218