A multi-dimensional resolution of singularities with applications to analysis

TL;DR

This paper introduces an explicit resolution of singularities algorithm for real-analytic functions in dimensions three and above, enabling precise analysis of zero sets, integrability, and oscillatory behavior.

Contribution

It develops a new elementary resolution algorithm inspired by algebraic geometry, applicable to real-analytic functions, and derives key analytical invariants and estimates.

Findings

Defined a new notion of the height of real-analytic functions

Computed the critical integrability index

Established the sharp growth rate of sublevel sets

Abstract

We formulate a resolution of singularities algorithm for analyzing the zero sets of real-analytic functions in dimensions $\geq 3$. Rather than using the celebrated result of Hironaka, the algorithm is modeled on a more explicit and elementary approach used in the contemporary algebraic geometry literature. As an application, we define a new notion of the height of real-analytic functions, compute the critical integrability index and obtain the sharp growth rate of sublevel sets. This also leads to a characterization of the oscillation index of scalar oscillatory integrals with real-analytic phases in all dimensions.