Zerofree region for exponenetial sums

TL;DR

This paper determines the minimal distance between the diagonal set and a specific exponential sum set in complex space, confirming a conjecture for large dimensions and providing an explicit formula.

Contribution

It proves that a particular point approximates the closest point on the exponential sum set to the diagonal for large n, confirming a conjecture about the minimal distance.

Findings

The minimal distance squared is approximately (\log n)^2 for large n.

The point (k, 0, ..., 0) with k = \log(n-1) + \\pi i is closest to the diagonal.

Explicit formula for the minimal distance involving the complex logarithm and the dimension n.

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 $d^2 = |k|^2 (1-1/n)$.