Recovery from Power Sums

TL;DR

This paper investigates the problem of recovering collections of numbers from power sum evaluations across various regimes, revealing complex behaviors and deviations from classical bounds in algebraic geometry and norm-based recovery.

Contribution

It provides a comprehensive analysis of power sum maps in different regimes and settings, including deviations from Bézout bounds and recovery methods using p-norms.

Findings

Fibers and images of power sum maps vary across regimes.

Deviations from Bézout bound are observed in certain settings.

Recovery of vectors from length measurements by p-norms is demonstrated.

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 B\'ezout bound, and the recovery of vectors from length measurements by $p$-norms.