On the solutions to a power sum problem

Abstract

In a recent paper we proved that if (*)=\inf_{|z_k|=1}\max_{v=1,...,n^2-n} |\sum_{k=1}^n z_k^v|, then (*)=\sqrt{n-1} if n-1 is a prime power. We proved that a construction of Fabrykowski gives minimal systems (z_1,...,z_n) to this problem. The construction depends on the existence of perfect difference sets of order n-1. As an open problem we asked whether all solutions would arise from this construction. In this paper we show that this is true and in fact if there exist no perfect difference set of order n-1 (which by the prime power conjecture is true if n-1 is not a prime power), then we have the strict inequality (*)>\sqrt{n-1}.