Document Type

Article

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This item is available under a Creative Commons License for non-commercial use only

Disciplines

Pure mathematics, Applied mathematics

Abstract

The presented analysis determines several new bounds on the roots of the equation $a_n x^n + a_{n−1} x^{n−1} + · · · + a_0 = 0$ (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 $a_0, a_1, . . . , a_{n−1}$, 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 $B_1, B_2, . . . B_q$ are 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 $k < l \le n$.

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