A Multi-dimensional Resolution of Singularities with Applications to Analysis

We formulate a resolution of singularities algorithm for analyzing the zero sets of real-analytic functions in dimensions ≥ 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 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.