Geodesic
On Lucas-Sets for Vector-Valued Abstract Polynomials in K-inner Product Spaces
Canadian journal of mathematics, Tome 34 (1982) no. 4, pp. 832-852

Voir la notice de l'article provenant de la source Cambridge University Press

The fact that Rolle's theorem on critical points of a real differentiable function does not hold in general for analytic functions of a complex variable raises a natural question [7, p. 21] as to whether or not it can be generalized to polynomials, the simplest subclass of analytic functions. While attempting to answer this and other related problems in [4, 5, 6], Lucas proved that all the critical points of a nonconstant polynomial f lie in the convex hull of the set of zeros of f (see Theorem (6.1) in [7]). Walsh [12] has shown that Lucas’ theorem is equivalent to the following result [7, Theorem (6.2)], namely: Any convex circular region which contains all the zeros of a polynomial f also contains all the zeros of its derivative f′. 
Zaheer, Neyamat. On Lucas-Sets for Vector-Valued Abstract Polynomials in K-inner Product Spaces. Canadian journal of mathematics, Tome 34 (1982) no. 4, pp. 832-852. doi: 10.4153/CJM-1982-058-5
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