Multivariate Igusa theory: Decay rates of exponential sums

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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, yⱼₚ, 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|=(|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.