Multivariate Igusa theory: decay rates of exponential sums

Abstract

We obtain general estimates for exponential integrals of the form Ef(y)=∫ℤpnψ(∑j=1ryjfj(x))|dx|⁠, 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 |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 with some elementarily obtained explicit bounds for Ef with fj polynomials.