Premium
Elementary inequalities involving the roots of a polynomial with applications in harmonic analysis and number theory
Author(s) -
Kowalski Michael W.,
Wright James
Publication year - 2012
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jds029
Subject(s) - mathematics , congruence relation , polynomial , euclidean geometry , pure mathematics , harmonic , exponential function , field (mathematics) , algebra over a field , mathematical analysis , geometry , physics , quantum mechanics
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.