Gauss–Lucas theorem

The Gauss -Lucas is a relationship between the zeros of a polynomial P and its derivative P ' on. The amount of the roots of a polynomial is a set of points in the complex plane. The theorem shows that the zeros of the derivative P ' lie in the convex hull of the zeros of P. The Gauss -Lucas is named after Carl Friedrich Gauss and Félix Lucas.

The Gauss -Lucas

Let be a polynomial with complex coefficients and is the derivation of. Then all the zeros of lie in the convex hull of the zeros of.

History

The sentence was written down for the first time by Carl Friedrich Gauss in 1836, but not proved until 1879 by Félix Lucas.

Stronger statement

The zeros of P ' lie even in the convex hull of the points

And with j ≠ k, where the n roots of P.