Volume Of Sub-level Sets Of Homogeneous Polynomials

TL;DR

This paper presents a method to approximate the volume of sub-level sets of homogeneous polynomials using a sequence of generalized eigenvalue problems involving Hankel matrices, with entries computed in closed form.

Contribution

It introduces a novel approach to estimate volumes of polynomial sub-level sets through eigenvalue problems, enabling precise approximations with increasing matrix size.

Findings

Volume approximation converges to the true value as matrix size increases.

Entries of the Hankel matrices are explicitly computed in closed form.

The method provides a practical computational approach for polynomial sub-level set volumes.

Abstract

Consider the sub level set K := {x : g(x) $\le$ 1} where g is a positive and homogeneous polynomial. We show that its Lebesgue volume can be approximated as closely as desired by solving a sequence of generalized eigenvalue problems with respect to a pair of Hankel matrices of increasing size, and whose entries are obtained in closed form.