Zerofree region for exponenetial sums

Abstract

We consider the following two closed sets in $C^n$. One is the diagonal D given by $(z,z,z,...z_{)}$. The other is $A=\{(z_1,z_2,z_3,...z_n):.$ $.e^{z_1}+e^{z_2}+e^{z_3}+...+e^{z_n}=0\}$. Clearly $D\cap A$ is empty. One can ask what is the distance between them. In this connection, Stolarsky [1] proved that the distance $d$ is given by $d^2=(\log n{)}^2+O(1)$. Some simple calculations will make one believe that the point $(k,0,0,..0)$ with $k=\log (n-1)+\pi i$ which lies on $A$ is one of the closest point to the diagaonal. We prove that this is indeed the case, atleast for sufficiently large $n$. This gives $d^2=|k{|}^2(1-1/n)$.