Improvements of Lagrange's bound for polynomial roots

An 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.