Multivariate Igusa theory: Decay rates of exponential sums

Abstract

We obtain general estimates for exponential integrals of the form \[ E_f(y)=\int_{\mathbb{Z}_{p}^{n}}ψ(\sum_{j=1}^r y_j f_j(x))|dx|, \] where the $f_j$ are restricted power series over ${\mathbb{Q}}_p$, $y_j\in {\mathbb{Q}}_p$, and $\psi$ a nontrivial additive character on ${\mathbb{Q}}_p$. We prove that if $(f_1,...,f_r)$ is a dominant map, then $|E_f(y)|<c|y{|}^{\alpha }$ for some $c>0$ and $\alpha <0$, uniform in $y$, where $|y|=\max (|y_i|{)}_i$. 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 $E_f$ with $f_j$ polynomials.

Improved results and presentation