Two theorems about the geometry of the critical points of a complex polynomial

Manuel Eberl 📧

November 14, 2023


This entry formalises two well-known results about the geometric relation between the roots of a complex polynomial and its critical points, i.e. the roots of its derivative.

The first of these is the Gauß–Lucas Theorem: The critical points of a complex polynomial lie inside the convex hull of its roots.

The second one is Jensen's Theorem: Every non-real critical point of a real polynomial lies inside a disc between two conjugate roots. These discs are called the Jensen discs: the Jensen disc of a pair of conjugate roots $a \pm b i$ is the smallest disc that contains both of them, i.e. the disc with centre $a$ and radius $b$.


BSD License


Session Polynomial_Crit_Geometry