Recovering an homogeneous polynomial from moments of its level set

TL;DR

This paper demonstrates that the moments of the Lebesgue measure on a sub-level set uniquely determine the defining homogeneous polynomial, allowing for its exact recovery via a simple linear system.

Contribution

It establishes that moments up to order 2d encode all information needed to recover a homogeneous polynomial defining a sub-level set.

Findings

Coefficients of the polynomial can be recovered from moments via a linear system.

Moments of order d and 2d are sufficient for polynomial reconstruction.

The associated matrix in the linear system is nonsingular.

Abstract

Let $K:={x: g(x)\leq 1}$ be the compact sub-level set of some homogeneous polynomial $g$. Assume that the only knowledge about $K$ is the degree of $g$ as well as the moments of the Lebesgue measure on $K$ up to order 2d. Then the vector of coefficients of $g$ is solution of a simple linear system whose associated matrix is nonsingular. In other words, the moments up to order 2d of the Lebesgue measure on $K$ encode all information on the homogeneous polynomial $g$ that defines $K$ (in fact, only moments of order $d$ and 2d are needed).