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A multi-dimensional resolution of singularities with applications to analysis
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 135, Number 5, October 2013
- pp. 1179-1252
- 10.1353/ajm.2013.0042
- Article
- Additional Information
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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 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.