New Bounds on the Real Polynomial Roots

Abstract

The presented 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. The following bounds are also found in this paper: max$\{ 1, ( \sum_{j = 1}^{q} B_j/A_l )^{1/(l-k)}\}$, where $B_1, B_2, \ldots B_q$ are the absolute values of all of the negative coefficients in the equation, $k$ is the highest degree of a monomial with a negative coefficient, $A_l$ is the positive coefficient of the term $A_l x^l$ for which $k< l \le n$.