2003/06/24 by Raf Cluckers, Cluckers, Raf
Mathematics · #03C10 #11L05 #11L07 #11S80 #11U09 #32B20 #32P05 #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Logic (math.LO) #Number Theory (math.NT) #Random Matrices and Applications #math.LO #math.NT #msc:03C10 #msc:11L05 #msc:11L07 #msc:11S80 #msc:11U09 #msc:32B20 #msc:32P05
paper · pdf · doi:10.48550/arxiv.math/0306351
Improved results and presentation
openalex publication_date 2003/06/24 · arxiv created 2004/08/10 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We obtain general estimates for exponential integrals of the form Ef(y)=∫_ℤpnψ(∑j=1r yj fj(x))|dx|, where the fj are restricted power series over ℚp, yj∈ℚp, and ψ a nontrivial additive character on ℚp. We prove that if (f1,...,fr) is a dominant map, then |Ef(y)| < c|y|α for some c>0 and α<0, uniform in y, where |y|=max(|yi|)i. In fact, we obtain similar estimates for a much bigger class of exponential integrals. To prove these estimates we introduce a new method to study exponential sums, namely, we use the theory of p-adic subanalytic sets and p-adic integration techniques based on p-adic cell decomposition. We compare our results to some elementarily obtained explicit bounds for Ef with fj polynomials.