Newton polyhedra and zeros of systems of exponential sums
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.
scientific article; zbMATH DE number 3953218