Multivariate Igusa theory: decay rates of exponential sums

We obtain general estimates for exponential integrals of the form $E_f{(y)}=\int_{\mathbb{Z}_p^n}\psi{(\sum\nolimits_{j=1}^ry_jf_j{(x)})}{\vert dx\vert}$ , where the fj are restricted power series over ℚp, yj ε ℚp and ψ is a nontrivial additive character on ℚp. We prove that if (f1,…,fr) is a dominant map, then ${\vert E_f{(y)}\vert}<c{\vert y\vert}^\alpha$ for some c > 0 and α < 0, uniform in y, where $\vert y\vert=\operatorname{max}(\vert y_i\vert)_i$ . 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 with some elementarily obtained explicit bounds for Ef with fj polynomials.