Turan幂和理论下一类inf max问题的显式解
Explicit solutions to certain inf max problems from Turan power sum theory
摘要：
设s_v表示纯幂和\sum_{k=1}^n z_k^v。在以前的文章中，我们证明了当n+1是素数时，\sqrt n <= \inf_{|z_k| => 1} \max_{v=1,...,n^2} |s_v| <= \sqrt{n+1}。本文证明了当n-1为素数幂时，\inf_{|z_k| = 1} \max_{v=1,...,n^2-n} |s_v| = \sqrt{n-1}，当2 <= i <= n-1且n => 3为素数幂时，\inf_{|z_k| => 1} \max_{v=1,...,n^2-i} |s_v| = \sqrt n.我们给出了n元组（z_1，.，z_n）的全局极小值。这是两个少数几次在图兰权力和理论的解决方案在inf最大问题可以明确计算。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
