A formula for separating small roots of a polynomial
Tateaki Sasaki, Akira Terui · ACM SIGSAM Bulletin · 2002
Let P(x) be a univariate polynomial over C, such that P(x) = c n x n + ... + c m+1 x m+1 + x m + e m-1 x m-1 + ... + e 0 , where max{ |c n |, ..., |c m+1 | } = 1 and e = max{ |e m-1 |, |e m-2 | 1/2 , ..., |e 0 | 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 D in of radius R in and other n - m roots outside a disc D out of radius R out , located at the origin, where R in(out) = [1 - (+) √1 - (16 e )/(1 + 3 e ) 2 ] × (1 + 3 e )/4. Note that R in = R out 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.