Dissemination of zeros of generalized derivative of a polynomial via matrix approach

Abstract

A polynomial of degree \(n\) is denoted by \(p(z):=\sum\limits_{j=0}^{n} a_{j}z^{j}\). Then, by classical Cauchy's result, $$|z|\le 1+ \max\bigg(\left|\frac {a_{n-1}}{a_{n}}\right|, \left|\frac {a_{n-2}}{a_{n}}\right|, \left|\frac {a_{n-3}}{a_{n}}\right|,...,\left|\frac {a_{0}}{a_{n}}\right|\bigg)$$ contains all of \(p(z)\)'s zeros. In order to improve on classical Cauchy finding, we will extend such results in this study to the polar derivative of an algebraic polynomial using matrix technique.