Abstract

We study the problem of recovering a collection of n numbers from the evaluation of m power sums. This yields a system of polynomial equations, which can be underconstrained (m < n), square (m = n), or overconstrained (m > n). Fibers and images of power sum maps are explored in all three regimes, and in settings that range from complex and projective to real and positive. This involves surprising deviations from the Bézout bound, and the recovery of vectors from length measurements by p-norms.

中文翻译：

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从功率和恢复

摘要

我们研究了从m个幂和的评估中恢复n 个数字的集合的问题。这会产生一个多项式方程组，它可以是欠约束 ( m < n )、平方 ( m = n ) 或过约束 ( m > n )。在所有三种情况下以及从复杂和投影到真实和积极的设置中，都探索了功率和图的纤维和图像。这涉及到 Bézout 界的惊人偏差，以及通过p范数从长度测量中恢复向量。