Coloring,sparseness and girth

Using the classical analysis resolution of singularities algorithm of G4], we generalize the theorems of G3] on Rⁿ sublevel set volumes and oscillatory integrals with real phase function to functions over an arbitrary local field of characteristic zero. 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. 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 maximum order of the zeroes of certain polynomials corresponding to the compact faces of this Newton polyhedron.