Estimations of the Bounds for the Zeros of Polynomials Using Matrices

Abstract

Let $$p\left( z\right) =z^{n}+\alpha _{n}z^{n-1}+\alpha _{n-2}z^{n-2}+\cdots +\alpha _{2}z+\alpha _{1}$$ be a monic polynomial of degree $$n\ge 7$$ with complex coefficients $$\alpha _{n},\alpha _{n-1},\ldots .\alpha _{1}$$ , where $$\alpha _{1}\ne 0$$ . This paper investigates and estimates the upper bounds for the moduli of the zeros of p depending on the spectral norms, spectral radii, and the fifth power of the Frobenius companion. These upper bounds allow us to locate all the zeros of p in smaller annuli in the complex plane.