2011/04/25 by Michael Greenblatt, Greenblatt, Michael
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical and Theoretical Analysis #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1104.4684
openalex publication_date 2011/04/25 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We use the resolution of singularities algorithm of [G4] to provide new\nestimates for exponential sums as well as new bounds on how often a function\nf(x) such as a polynomial with integer coefficients is divisible by various\npowers of a prime p when x is an integer. They are proved using p-adic\nanalogues of the theorems of [G3] on Rn sublevel set volumes and oscillatory\nintegrals with real phase function. The proofs of these analogues use aspects\nof the resolution of singularities algorithms of [G4] (but for the most part\nnot the actual resolution of singularities theorems themselves.)\n Unlike many papers on such exponential sums and p-adic oscillatory integrals,\nwe do not require the Newton polyhedron of the phase to be nondegenerate, but\nrather as in [G3] we have conditions on the maximal order of the zeroes of\ncertain polynomials corresponding to the compact faces of the Newton polyhedron\nof the phase function.\n