Mathematics – Classical Analysis and ODEs
Scientific paper
2011-04-25
Mathematics
Classical Analysis and ODEs
40 pages. v3: various clarifications and corrections
Scientific paper
In this paper, the author's earlier elementary local resolution of singularities algorithm [G1]-[G3] is simplified and extended to functions with convergent power series expansions over a general local field of characteristic zero. Furthermore, the theorems of [G3] on R^n sublevel set volumes and oscillatory integrals with real phase function are generalized to such functions. The p-adic cases of these results immediately imply 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 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 maximal order of the zeroes of certain polynomials corresponding to the compact faces of the Newton polyhedron of the phase function.
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