New Bounds on the Real Polynomial Roots

Authors

  • Emil M. Prodanov School of Mathematical Sciences, Technological University - Dublin

DOI:

https://doi.org/10.7546/CRABS.2022.02.02

Keywords:

polynomial equation, root bounds, Cauchy polynomial, Cauchy theorem, Cauchy and Lagrange bounds, Descartes’ rule of signs

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 $$k<l\le n$$.

Author Biography

Emil M. Prodanov, School of Mathematical Sciences, Technological University - Dublin

School of Mathematical Sciences, Technological University - Dublin
Park House, Grangegorman
191 North Circular Road Dublin D07 EWV4, Ireland
e-mail: emil.prodanov@tudublin.ie

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Published

02-03-2022

How to Cite

[1]
E. Prodanov, “New Bounds on the Real Polynomial Roots”, C. R. Acad. Bulg. Sci. , vol. 75, no. 2, pp. 178–186, Mar. 2022.

Issue

Section

Mathematics