Multivariate Igusa theory: Decay rates of exponential sums

Abstract

We obtain general estimates for exponential integrals of the form \[ E_f(y)=\int_{\mathbb{Z}_{p}^{n}}\psi(\sum_{j=1}^r y_j f_j(x))|dx|, \] 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.