On the critical points of a polynomial

I. Nazir, N. Ahmed dar, M. I. Mir

The classical Gauss-Lucas Theorem states that all the critical points of a non -constant polynomial f(z) lie in the convex hull K(f) of the zeros of f(z). The present work provides a refinement of this classical result by giving a more precise localization of the critical points of second kind, i.e., those critical points that are not zeros of f(z). We also prove a theorem which will give a disk in which at least one critical point of f(z) lies and the result is best possible. 

Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 19(3) (2026), pp. 11-18