Abstract

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 ± bi is the smallest disc that contains both of them, i.e. the disc with centre a and radius b.

BibTeX

@article{Polynomial_Crit_Geometry-AFP,
  author  = {Manuel Eberl},
  title   = {Two theorems about the geometry of the critical points of a complex polynomial},
  journal = {Archive of Formal Proofs},
  month   = {November},
  year    = {2023},
  note    = {\url{https://isa-afp.org/entries/Polynomial_Crit_Geometry.html},
             Formal proof development},
  ISSN    = {2150-914x},
}