New Bounds on the Real Polynomial Roots

Abstract

The presented analysis determines several new bounds on the real roots of the equation $$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0=0$$ (with $$a_n>0$$). All proposed new bounds are lower than the Cauchy bound $$\max\left\{1,\sum_{j=0}^{n-1}|a_j/a_n|\right\}$$. 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 $$a_0,a_1,\dots,a_{n-1}$$, 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: $$\max\left\{1,\left(\sum_{j=1}^{q} B_j/A_l\right)^{1/(l-k)}\right\}$$, where $$B_1,B_2,\dots,B_q$$ 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 $$k<l\le n$$.