Applications of an elementary resolution of singularities algorithm

In this paper, we use the resolution of singularities algorithm of [G4] to generalize to arbitrary local fields of characteristic zero the theorems of [G3] on R sublevel set volumes and oscillatory integrals with real phase function. The proofs of these generalizations use various aspects of the resolution of singularities algorithms of [G4] (but for the most part not the actual resolution of singularities theorems themselves.) The p-adic cases of our results provide new estimates for exponential sums as well as new bounds on how often a function f(x) such as a polynomial with integer coefficients is divisible by various powers of a prime p when x is an integer. Thus we use classical analysis resolution of singularities methods on a class of problems traditionally approached using toric resolution of singularities techniques.