Batra, PrashantPrashantBatraMignotte, MauriceMauriceMignotteŞtefănescu, DoruDoruŞtefănescu2019-11-222019-11-222017-09-01Journal of Symbolic Computation (82): 19-25 (2017-09-01)http://hdl.handle.net/11420/3857An upper bound for the roots of Xd+a1Xd−1+⋯+ad−1X+ad is given by the sum of the largest two of the terms |ai|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.en0747-717120171925Complex rootsComputational effortOverestimation factorJournal Article10.1016/j.jsc.2016.10.001Other