Explicit solutions to certain inf max problems from Turan power sum theory

Abstract

Let s_v denote the pure power sum \sum_{k=1}^n z_k^v. In a previous paper we proved that \sqrt n <= \inf_{|z_k| => 1} \max_{v=1,...,n^2} |s_v| <= \sqrt{n+1} when n+1 is prime. In this paper we prove that \inf_{|z_k| = 1} \max_{v=1,...,n^2-n} |s_v| = \sqrt{n-1} when n-1 is a prime power, and if 2 <= i <= n-1 and n => 3 is a prime power then \inf_{|z_k| => 1} \max_{v=1,...,n^2-i} |s_v| =\sqrt n. We give explicit constructions of n-tuples (z_1,...,z_n) which we prove are global minima for these problems. These are two of the few times in Turan power sum theory where solutions in the inf max problem can be explicitly calculated.