Treffer: A generalization of Lucas′ theorem to vector spaces: A generalization of Lucas' theorem to vector spaces

Let \(E\) and \(V\) denote vector spaces over an algebraically closed field of characteristic zero and \({\mathcal P}^*\) denote the family of all nonconstant polynomials \(P:E\to V\). The concept of Lucas-sets for the family \({\mathcal P}^*\), when \(E\) is a \(K\)-inner product space, was introduced by \textit{N. Zaheer} [Can. J. Math. 34, 832-852 (1982; Zbl 0451.46015)] and it was shown that every member \(A\) of the family \(D(E_ \omega)\) of all generalized circular regions of \(E_ \omega\), with \(\omega\notin A\), is a Lucas-set for \({\mathcal P}^*\). This fact raises two questions: Firstly, does \(D(E_ \omega)\) exhaust all Lucas-sets in \(E_ \omega\) when \(E\) is a \(K\)-inner product space? Secondly, does there exists an analogous family of Lucas-sets for \({\mathcal P}^*\) when \(E\) is, in general, a vector space? In this paper the author introduced the family \(D^*(E_ \omega)\) of super-generalized circular region of \(E_ \omega\) which answers the first question negatively and the second question affirmatively. This family was employed to generalize the classical Lucas' theorem on the zeros of the derivative of a polynomial to vector-valued abstract polynomials in vector spaces.