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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Newton polygons and local integrability of negative powers of smooth functions in the plane


Author: Michael Greenblatt
Journal: Trans. Amer. Math. Soc. 358 (2006), 657-670
MSC (2000): Primary 42B20; Secondary 35H20
Published electronically: February 4, 2005
MathSciNet review: 2177034
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Abstract | References | Similar Articles | Additional Information

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

\begin{displaymath}\sup\left\{\epsilon:\int_U \vert f(x,y)\vert^{-\epsilon}<\infty\right\}. \end{displaymath}

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.


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Additional Information

Michael Greenblatt
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03664-0
PII: S 0002-9947(05)03664-0
Keywords: Resolution of singularities, Newton polygon
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
Article copyright: © Copyright 2005 American Mathematical Society