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

 

Exponential sums: Questions by Denef, Sperber, and Igusa


Author: Raf Cluckers
Journal: Trans. Amer. Math. Soc. 362 (2010), 3745-3756
MSC (2010): Primary 11L07, 11S40; Secondary 11L05
Published electronically: December 3, 2009
MathSciNet review: 2601607
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Abstract: We prove the remaining part of the conjecture by Denef and Sperber [Denef, J. and Sperber, S., Exponential sums mod $ p^n$ and Newton polyhedra, Bull. Belg. Math. Soc., suppl. (2001) 55-63] on nondegenerate local exponential sums modulo $ p^m$. We generalize Igusa's conjecture in the introduction of [Igusa, J., Lectures on forms of higher degree, Lect. Math. Phys., Springer-Verlag, 59 (1978)] from the homogeneous to the quasi-homogeneous case and prove the nondegenerate case as well as the modulo $ p$ case. We generalize some results by Katz in [Katz, N. M., Estimates for ``singular'' exponential sums, Internat. Math. Res. Notices (1999) no. 16, 875-899] on finite field exponential sums to the quasi-homogeneous case.


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Additional Information

Raf Cluckers
Affiliation: Departement wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
Address at time of publication: Laboratoire Painlevé, Université Lille 1, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex France
Email: raf.cluckers@wis.kuleuven.be

DOI: http://dx.doi.org/10.1090/S0002-9947-09-05084-3
PII: S 0002-9947(09)05084-3
Keywords: Exponential sums, nondegenerate polynomials, Igusa's conjecture on exponential sums, Igusa's local zeta functions, motivic integration
Received by editor(s): September 4, 2008
Published electronically: December 3, 2009
Additional Notes: The author was a postdoctoral fellow of the Fund for Scientific Research - Flanders (Belgium) (F.W.O.)
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.