Exponential sums: questions by Denef, Sperber, and Igusa

2007/11/21 by Raf Cluckers, Cluckers, R.
Computer Science · Mathematics · #11L05 #11L07 #11S40 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0711.3365

openalex publication_date 2007/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the remaining part of the conjecture by Denef and Sperber [Denef, J. and Sperber, S., \textitExponential sums mod pⁿ and Newton polyhedra, Bull. Belg. Math. Soc., \bfsuppl. (2001) 55-63] on nondegenerate local exponential sums modulo p^m. We generalize Igusa's conjecture of the introduction of [Igusa, J., Lectures on forms of higher degree, Lect. math. phys., Springer-Verlag, \bf59 (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 of [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.

Citations

• Igusa and Denef-Sperber Conjectures on nondegenerate p-adic exponential sums

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