Newton polygons and local integrability of negative powers of smooth functions in the plane
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- by Michael Greenblatt PDF
- Trans. Amer. Math. Soc. 358 (2006), 657-670 Request permission
Abstract:
Let $f(x,y)$ be any smooth real-valued function with $f(0,0)=0$. For a sufficiently small neighborhood $U$ of the origin, we study the number \[ \sup \left \{\epsilon :\int _U |f(x,y)|^{-\epsilon }<\infty \right \}. \] It is known that sometimes this number can be expressed in a natural way using the Newton polygon of $f$. We provide necessary and sufficient conditions for this Newton polygon characterization to hold. The behavior of the integral at the supremal $\epsilon$ is also analyzed.References
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Additional Information
- Michael Greenblatt
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
- Received by editor(s): November 11, 2002
- Received by editor(s) in revised form: February 6, 2004
- Published electronically: February 4, 2005
- Additional Notes: This research was supported in part by NSF grant DMS-9988798
- © Copyright 2005 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 657-670
- MSC (2000): Primary 42B20; Secondary 35H20
- DOI: https://doi.org/10.1090/S0002-9947-05-03664-0
- MathSciNet review: 2177034