A theorem of Gauss

Quote:

Originally Posted by Bruno J.

Let $p(z)$ be a complex polynomial of degree $>1$ . Show that the zeroes of $p'(z)$ lie in the convex hull of the zeroes of $p(z)$ .

Spoiler:

This is Gauss-Lucas Theorem. Highlights of the proof:

If $p(z)=p'(z)=0$ , then $z=1\cdot z+\sum^n_{i=1}0\cdot a_1$ is a convex combination of the roots of $p(z)$ , otherwise:

write $p(z)=a\prod\limits^n_{i=1}(z-a_i) \,,\,\,a,a_i\in\mathbb{C}$ , and take the logarithmic derivative of this:

$\frac{p'(z)}{p(z)}=\sum^n_{i=1}\frac{1}{z-a_i}=\sum^n_{i=1}\frac{\overline{z}-\overline{a_i}}{|z-a_i|^2}$ , which is valid whenever $p(z)\neq 0$ . So if $z$ is a root of $p'(z)$ but not of $p(z)$ we get:

$0=\sum^n_{i=1}\frac{\overline{z}-\overline{a_i}}{|z-a_i|^2}\Longrightarrow \sum^n_{i=1}\frac{1}{|z-a_i|^2}\overline{z}=\sum^n_{i=1}\frac{1}{|z-a_i|^2}\overline{a_i}$ , and taking conjugates and dividing we finally get:

$z=\frac{\displaystyle{\sum^n_{i=1}\left(\frac{1}{| z-a_i|^2}\right)a_i}}{\displaystyle{\sum^n_{i=1}\fra c{1}{|z-a_i|^2}}}$ $= \frac{\displaystyle{\sum^n_{i=1}\frac{1}{|z-a_i|^2}}}{\displaystyle{\sum^n_{k=1}\frac{1}{|z-a_k|^2}}}\,a_i$ , and it's easy now to see these are barycentric coordinates of $z$ wrt $a_1,\ldots,a_n$ (please do pay attention

closely to the last expression's indexes).

Tonio