On the closure of the real parts of the zeros of a class of exponential polynomials

Abstract

In this paper we give a characterization of the set RPN(z):={Rz:PN(z)=0}, in terms of the position of a line with respect to an analytic variety, with PN(z) belonging to a large class P(z) of exponential polynomials which can be expressed of the form 1+∑Nj=1aje−zγj.r, where N, M are positive integers, aj∈R with aj≠0, γj=(γj1,γj2,…,γjM), j=1,…,N, are non- null vectors, distinct, with non-negative integers components, r=(r1,r2,…,rM) is a vector of RM with positive rationally independent components, γj.r is the inner product of γj by r in RM, and, for some 1≤jN≤N, is γjN.γj=0 , for all j≠jN.