Volume estimates of sublevel sets of real polynomials

TL;DR

This paper provides upper bounds for the volume of sublevel sets of real polynomials by combining Lojasiewicz inequalities with volume estimates, with applications to oscillatory integrals and polynomial indices.

Contribution

It introduces a novel method that combines global Lojasiewicz inequalities with volume estimates to bound sublevel set volumes of real polynomials.

Findings

Derived explicit upper bounds for sublevel set volumes

Applied bounds to oscillatory integrals

Connected volume estimates to polynomial integration indices

Abstract

We give upper bounds for volume of sublevel sets of real polynomials. Our method is to combine a version of global Lojasiewicz inequality with some well known estimate on volume of tubes around real algebraic sets. Some applications to oscillatory integrals and integration indices of real polynomial are also given.