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We use the resolution of singularities algorithm of [G4] to provide newestimates for exponential sums as well as new bounds on how often a functionf(x) such as a polynomial with integer coefficients is divisible by variouspowers of a prime p when x is an integer. They are proved using p-adicanalogues of the theorems of [G3] on R^n sublevel set volumes and oscillatoryintegrals with real phase function. The proofs of these analogues use aspectsof the resolution of singularities algorithms of [G4] (but for the most partnot the actual resolution of singularities theorems themselves.)  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, butrather as in [G3] we have conditions on the maximal order of the zeroes ofcertain polynomials corresponding to the compact faces of the Newton polyhedronof the phase function.

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