A multi-dimensional resolution of singularities with applications to\n analysis

2010/07/03 by Tristan C. Collins, Collins, Tristan, Allan Greenleaf +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Advanced Differential Equations and Dynamical Systems #Black Holes and Theoretical Physics

paper · pdf · doi:10.48550/arxiv.1007.0519

Abstract

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

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