Improvements of Lagrange's bound for polynomial roots. (September 2017) - Batra, Prashant; Mignotte, Maurice; Ştefănescu, Doru
Abstract: An upper bound for the roots of X d + a 1 X d − 1 + ⋯ + a d − 1 X + a d is given by the sum of the largest two of the terms | a i | 1 / i . This bound by Lagrange has gained attention from different sides recently, while a succinct proof seems to be missing. We present a short, original proof of Lagrange's bound. Our approach leads to some definite improvements. To benefit computationally from these improvements, we construct a modified Lagrange bound which at the same asymptotic computational complexity is at most 11 per cent from optimal for degrees d ≥ 16 .