Gauss-Lucas theorem

The Gauss-Lucas mathematical theorem gives a relationship between the zeros of a polynomial and its derivative . The set of roots of a polynomial is a set of points in the complex plane . The theorem shows that the roots of the derivative lie in the convex hull of the roots of . The Gauss-Lucas theorem is named after Carl Friedrich Gauß and Félix Lucas . ${\displaystyle P}$ ${\displaystyle P'}$ ${\displaystyle P'}$ ${\displaystyle P}$

The Gauss-Lucas theorem

Let be a polynomial function with complex coefficients and be the derivative of . Then all zeros of lie in the convex hull of the zeros of . ${\displaystyle P}$ ${\displaystyle P^{\prime }}$ ${\displaystyle P}$ ${\displaystyle P^{\prime }}$ ${\displaystyle P}$

history

The sentence was first written down by Carl Friedrich Gauß in 1836, but only proved by Félix Lucas in 1879 .

Stronger statement

The zeros of are even in the convex hull of the points ${\displaystyle P'}$

${\displaystyle \omega _{jk}={\frac {z_{j}+(n-1)z_{k}}{n}}}$

with and , where the zeros are from. ${\displaystyle j\,,k=1,\ldots ,n}$ ${\displaystyle j\neq k}$ ${\displaystyle z_{1},\ldots ,z_{n}}$ ${\displaystyle n}$ ${\displaystyle P}$

Individual evidence

1. ↑ CF Gauß: Werke , Volume 3 , Göttingen 1866, p. 120: 112
2. ^ F. Lucas: Sur une application de la Mécanique rationnelle à la théorie des equations. in: Comptes Rendus de l'Académie des Sciences (89), Paris 1979, pp. 224-226
3. ↑ W. Specht: A comment on the sentence by Gauß-Lucas , in: Annual report of the German Mathematicians Association (62), 1959, pp. 85-92

See also

• Marden's theorem

Web links

• Lucas – Gauss theorem by Bruce Torrence, of the Wolfram Demonstrations Project.