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

TL;DR

This paper applies an elementary resolution of singularities algorithm to derive new estimates for exponential sums and bounds on divisibility of polynomial functions by prime powers, using p-adic analogues of oscillatory integral theorems.

Contribution

It introduces a novel approach that avoids the need for nondegeneracy conditions on the Newton polyhedron, expanding the applicability of these estimates.

Findings

New bounds on exponential sums and divisibility by prime powers.

P-adic analogues of oscillatory integral theorems without nondegeneracy assumptions.

Application of resolution of singularities to p-adic exponential sums.

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] (but for the most part not 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, 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.