Certain compositions of abstract polynomials and their pseudo- derivatives (Q910517)

If E is a linear space over an algebraically closed field K then a map P: \(E\to K\) is called an abstract polynomial, if for all x, y in E and c in K one has \[ P(x+cy)=\sum^{n}_{k=0}A_ k(x,y)c^ k, \] where \(A_ k(x,y)\) are elements of K, independent of c. The authors consider pseudo- derivatives of abstract polynomials and generalize certain results of \textit{N. Zaheer} and \textit{M. Alam} [Proc. Lond. Math. Soc., III. Ser. 40, 527-552 (1980; Zbl 0432.12012)] concerning zero-sets of \(f(z)-tf_ 1(w,z)\) and \(f(z)+t(z-w)f'(z)\), where t, w are fixed complex numbers and \[ f_ 1(w,z)=nf(z)+(w-z)f'(z) \] is the polar-derivative of the polynomial f, n being the degree of f.