Gauss–Lucas theorem

For every nonconstant complex polynomial, each zero of its derivative lies in the convex hull of the polynomial’s zeros, counted without regard to multiplicity.

Logarithmic derivative: The derivative of a function divided by the function itself, wherever the function is nonzero. At a derivative zero away from the roots, its root sum vanishes, yielding the convex-hull result.
Complex plane: The plane whose points represent complex numbers. Polynomial roots and their convex hull are interpreted as points and regions in this plane.
Carl Friedrich Gauss: A German mathematician whose work shaped number theory, analysis, geometry, and astronomy. The theorem bears his name and belongs to the classical study of polynomial roots.
Rolle's theorem: A theorem stating that a differentiable real function with equal endpoint values has a stationary point between them. It is the real-variable counterpart to the derivative-root location phenomenon.
Coincident zeros of a polynomial and its derivative: A shared zero of a polynomial and its derivative, corresponding to a repeated root. Repeated roots lie on the hull boundary and expose an equality case of the theorem.
Convex combination: A weighted sum of points with nonnegative weights that add to one. The root-balance equation expresses a critical point as a convex combination of roots.
Fundamental theorem of algebra: Every nonconstant complex polynomial has a complex root and factors completely into linear factors. Its factorization into roots supplies the representation used in the theorem’s proof.
Isaac Lucas: A French mathematician known for work on number theory and polynomial equations. His name is attached to the theorem alongside Gauss’s.
Hermite–Biehler theorem: A criterion relating the real zeros of two polynomials to the stability of a complex polynomial. It belongs to the broader toolkit for locating polynomial zeros and assessing stability.
Gauss–Lucas theorem for rational functions: Extensions of the polynomial result that constrain critical points of rational functions using their zeros and poles. Generalizing the root-only convex-hull statement requires accounting for poles.

Linked from 3 pages