A formula for separating small roots of a polynomial

Abstract

Let P(x) be a univariate polynomial over C, such that P(x) = cnxⁿ + ... + cm+1x^m+1 + x^m + em-1x^m-1 + ... + e0, where max{|cn|, ..., |cm+1|} = 1 and e = max{|em-1|, |em-2|^1/2, ..., |e0|^1/m} << 1. P(x) has m small roots around the origin so long as e << 1. In 1999, we derived a formula that if e < 1/9 then P(x) has m roots inside a disc Din of radius Rin and other n - m roots outside a disc Dout of radius Rout, located at the origin, where Rin(out) = [1 - (+) √1 - (16e)/(1 + 3e)²] × (1 + 3e)/4. Note that Rin = Rout if e = 1/9. Our formula is essentially the same as that derived independently by Yakoubsohn at almost the same time. In this short article, we introduce the formula and check its sharpness on many polynomials generated randomly.