On pseudo polar-derivatives of abstract polynomials

Abstract

We introduce the concept of pseudo polar-derivatives of abstract polynomials from E to K ( E being a vector space over an algebraically closed field K of characteristic zero) which generalizes to vector spaces the well-known concept of polar derivatives (cf. [4, pp. 44 and 52]) for ordinary polynomials from C to C (the field of complex numbers). Herein we employ supergeneralized circular regions, a vector space analogue of classical circular regions in the complex plane, to determine the location of null-sets of pseudo polar-derivatives of abstract polynomials. Our main theorem of this paper generalizes the corresponding classical version due to Laguerre [4, Theorems (13, 1)–(13, 2)] and a field analogue due to Zervos [9, Corollary (4.2), p. 360]. Some interesting examples are also discussed to throw light on some relevant facts.