Recovering an Homogeneous Polynomial from Moments of Its Level Set

Abstract

Let $$\mathbf{K }:=\left\{ \mathbf{x }: g(\mathbf{x })\le 1\right\} $$K:=x:g(x)≤1 be the compact (and not necessarily convex) sub-level set of some homogeneous polynomial $$g$$g. Assume that the only knowledge about $$\mathbf{K }$$K is the degree of $$g$$g as well as the moments of the Lebesgue measure on $$\mathbf{K }$$K up to order $$2d$$2d. Then the vector of coefficients of $$g$$g is the solution of a simple linear system whose associated matrix is nonsingular. In other words, the moments up to order $$2d$$2d of the Lebesgue measure on $$\mathbf{K }$$K encode all information on the homogeneous polynomial $$g$$g that defines $$\mathbf{K }$$K (in fact, only moments of order $$d$$d and $$2d$$2d are needed).