Elementary inequalities involving the roots of a polynomial with applications in harmonic analysis and number theory

Abstract

We establish various inequalities relating the coefficients of a polynomial with the separation of its roots. Applications are given to oscillatory integrals and sublevel sets in euclidean harmonic analysis as well as exponential sums and polynomial congruences in number theory. These applications depend on precise structural statements of sublevel sets for polynomials with coefficients in a general field and these in turn give sharpened versions of classical results of Hua as well as Loxton and Smith regarding polynomial congruences.