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arXiv · 1007.0519

A multi-dimensional resolution of singularities with applications to analysis

Tristan Collins·Allan Greenleaf·Malabika Pramanik
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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.

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Tristan Collins, Allan Greenleaf, Malabika Pramanik. 2011-08-05. A multi-dimensional resolution of singularities with applications to analysis. https://arxiv.org/abs/1007.0519

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