Annular bounds for the zeros of a polynomial from companion matrices

This paper is concerned with the problem to locate the zeros of a polynomial by using matrix inequalities involving the spectral radius, the numerical radius and the spectral norm. This classical problem attracted many mathematicians over the years beginning with Cauchy. This problem is still an enchanting topic to both complex and numerical analysts. One can compute the zeros of a polynomial using the coefficients and their radicals whenever the degree of the polynomial is less than or equal to 4, however for degree greater than or equal to 5 this computation is not always possible. So the study of location of the zeros of a polynomial becomes interesting and useful for higher degree poynomials. The location of the zeros of polynomials have important applications in many areas of sciences such as signal processing, control theory, communication theory, coding theory and cryptography etc. The Frobenius companion matrix plays an important link between matrix theory and the geometry of polynomials. It has been used to obtain estimations for zeros of polynomials by matrix methods, we refer to some of the recent papers [3, 4, 5, 13] and the references therein. Also, various mathematicians have obtained annular regions containing all the zeros of a polynomial by using classical approach, we refer to [6, 7, 15] and references therein. Suppose that