A multi-dimensional resolution of singularities with applications to analysis (Q2856608)

The article presents an algorithm for resolving singularities of a real-analytic function in three or more variables. The motivation is threefold: (i) study the critical integrability index of a real-analytic function \(F\) at a point \(a\in F^{-1}(\{0\})\), that is, the supremum of real numbers \(\mu>0\) such that \(| F|^{-\mu}\) is integrable in a neighborhood of \(a\), (ii) estimate the so-called growth rate of sublevel sets of \(F\), and (iii) describe the oscillation index of a class of scalar oscillatory integrals with real-analytic phase. These problems are of importance in a number of questions in harmonic analysis, PDE, and geometry. The algorithm is based on an elementary and explicit approach using tools from local analytic geometry like the Abhyankar-Jung theorem, Newton polyhedra, fractional power series and multivariate Puiseux expansions, which allows effective computation, as well as several extremal characterizations, of all three aforementioned indices.