Gauss-Lucas Theorem
Published by Angel on Sunday, February 28, 2010 - 22:29:55 - Filed under Convex Sets
Im back with a very known theorem, I like it a lot, even if the proof is already on the Wikipedia Id like to write it here with an example because its a direct consequence of the Convex Hull post I wrote before.
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Gauss-Lucas Theorem:
The roots of the derivative of a non constant complex polynomial belong to the convex hull generated by the roots of the polynomial.
Proof:
Let 
Since we know that 
where z1,…,zn are the roots of the polynomial .
We can also write 
Now:
If z is a root of the derivative, then:
This means that:
Then :
Which means:
Then:
But with this we know that z is a convex combination of all the roots, this is because :
Then by this we know that z is in the convex hull of 
QED.
Example:
Proof that the diameter of the set of roots of the polynomial :
cannot be less than 
Proof:
We know that 
:
Now we just use that:
and 
Now its clear that:
.
Since the roots of the derivative are inside the convex hull generated by the roots of the original polynomial, then the diameter between them cannot be less than the diameter between the set of the roots of the second derivative of the polynomial.
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The example was one of the problems on my exam. Good luck I liked this theorem and I didnt forget about it :).
