How the Roots of a Polynomial Vary with its Coefficients: A Local Quantitative Result

Abstract

AbstractA well-known result, due to Ostrowski, states that if , then the roots (xj) of P and (yj) of Q satisfy , where n is the degree of P and Q. Though there are cases where this estimate is sharp, it can still be made more precise in general, in two ways: first by using Bombieri’s norm instead of the classical l1 or l2 norms, and second by taking into account the multiplicity of each root. For instance, if x is a simple root of P, we show that instead of . The proof uses the properties of Bombieri’s scalar product andWalsh Contraction Principle.