BOMBIERI'S NORM VERSUS MAHLER'S MEASURE

摘要

Factorization algorithms for polynomials with integer coefficients and one complex variable use an a priori bound on the size of the coefficients in any factor of P. The first bound of this type was given by Mignotte, using Mahler's measure. Then Beauzamy and Beauzamy-Trevisan-Wang gave sharper estimates, using Bombieri's norm. This leads to the natural question: for which polynomials is Bombieri's norm smaller than Mahler's measure? We give an answer here, in terms of the localization of the roots of P, more precisely a sufficient condition on the modulus of the roots, for a polynomial with complex coefficients and one complex variable to have its Bombieri's norm smaller than its Mahler's measure.