Exponential sums: questions by Denef, Sperber, and Igusa
Abstract
We prove the remaining part of the conjecture by Denef and Sperber [Denef, J. and Sperber, S., \textit{Exponential sums mod and {N}ewton polyhedra}, Bull. Belg. Math. Soc., {\bf{suppl.}} (2001) 55-63] on nondegenerate local exponential sums modulo . We generalize Igusa's conjecture of the introduction of [Igusa, J., \textit{Lectures on forms of higher degree}, Lect. math. phys., Springer-Verlag, {\bf{59}} (1978)] from the homogeneous to the quasi-homogeneous case and prove the nondegenerate case as well as the modulo case. We generalize some results by Katz of [Katz, N. M., \textit{Estimates for "singular" exponential sums}, Internat. Math. Res. Notices (1999) no. 16, 875-899] on finite field exponential sums to the quasi-homogeneous case.
Cite
@article{arxiv.0711.3365,
title = {Exponential sums: questions by Denef, Sperber, and Igusa},
author = {R. Cluckers},
journal= {arXiv preprint arXiv:0711.3365},
year = {2007}
}