New Bounds on the Real Polynomial Roots

Abstract

The presented two-tier analysis determines several new bounds on the roots of the equation \(a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 = 0\) (with \(a_n > 0\)). All proposed new bounds are lower than the Cauchy bound max\(\{1, \sum_{j=0}^{n-1} |a_j/a_n| \}\). Firstly, the Cauchy bound formula is derived by presenting it in a new light -- through a recursion. It is shown that this recursion could be exited at earlier stages and, the earlier the recursion is terminated, the lower the resulting root bound will be. Following a separate analysis, it is further demonstrated that a significantly lower root bound can be found if the summation in the Cauchy bound formula is made not over each one of the coefficients \(a_0, a_1, \ldots, a_{n-1}\), but only over the negative ones. The sharpest root bound in this line of analysis is shown to be the larger of 1 and the sum of the absolute values of all negative coefficients of the equation divided by the largest positive coefficient. Combining the two analyses, one can find a quadratic, a cubic, and a quartic equation the unique positive root of each of which provides a sharper bound than all other new bounds proposed in this paper.