### 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 $a \pm b i$ is the smallest disc that contains both of them, i.e. the disc with centre $a$ and radius $b$.

### License

BSD License

### Topics

- Mathematics/Algebra

### Session Polynomial_Crit_Geometry

- Polynomial_Crit_Geometry_Library
- Polynomial_Crit_Geometry

### Depends on

- Formal Puiseux Series

## Cite

×

Polynomial_Crit_Geometry-AFP

    @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},
    }

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