Zeros of certain composite polynomials in algebraically closed fields

Abstract

The study of the zeros of composite polynomials has mostly been confined to polynomials in the complex plane. The object of this paper is to study the zeros of the composite polynomials which arise as linear combinations of a polynomial and its (formal) derivatives in an algebraically closed field $K$ of characteristic zero. Our main theorem Concerning the zeros of such composite polynomials gives certain interesting results which, when applied to the complex plane, furnish improved versions of the corresponding classical results due to Walsh, Marden, and Kakeya. At the end we show that our results cannot be further generalized in certain directions.