Applications of an elementary resolution of singularities algorithm to exponential sums and congruences modulo p^n

Abstract: We use the resolution of singularities algorithm of [G4] to 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. They are proved using p-adic analogues of the theorems of [G3] on R^n sublevel set volumes and oscillatory integrals with real phase function. The proofs of these analogues use aspects of the resolution of singularities algorithms of G4 Unlike many papers on such exponential sums and p-adic oscillatory integrals, we do not require the Newton polyhedron of the phase to be nondegenerate, but rather as in [G3] we have conditions on the maximal order of the zeroes of certain polynomials corresponding to the compact faces of the Newton polyhedron of the phase function.