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[complex functions] polynomial roots in a convex hull

  • Thread starter rahl___
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Hi everyone,

I've got this problem to solve:

Let [tex]W(z)[/tex] be a polynomial with complex coefficients and complex roots. Show that the roots of [tex]W'(z)[/tex] are in a convex hull of the set of roots of [tex]W(z)[/tex].
My problem is that I don't fully understand the question.

I have found such definition of convex hull:
Given a set of points [tex](z_1, z_2, ..., z_n)[/tex], we denote convex hull as:
[tex]conv(z_1, z_2, ..., z_n) = \{ z = \sum_{k=1}^n \beta_k z_k : \beta_k \in [0,1], \sum_{k=1}^n \beta_k = 1 \}[/tex]
So I do have to prove, that all the roots of [tex]W'(z)[/tex] [let's denote them as [tex]z'_k[/tex]] must be able to be written in such form:
[tex]z'_k = \sum_{k=1}^n \beta_k z_k[/tex], where [tex]\beta_k[/tex] are satysfying the conditionsof convex hull and [tex]z_k[/tex] are the roots of [tex]W(z)[/tex].
Am i right?

If so, I thought about this kind of sollution:
we assume that what we have to prove is true, so we can write the roots of [tex]W'(z)[/tex] as:
[tex]z'_j = \sum_{k=1}^n \beta_k^{(j)} z_k[/tex]
Now we write down [tex]W'(z)[/tex] using viete's formulas:
[tex]W'(z) = n a_n z^{n-1} - n a_n ( \sum_j \sum_k \beta_k^{(j)} z_k ) z^_{n-2} + n a_n \sum_{i<j} ( \sum_k \beta_k^{(i)} z_k ) ( \sum_k \beta_k^{(j)} z_k ) z^{n-3} - ... + n a_n \prod_j \sum_k \beta_k^{(j)} z_k[/tex] [dont know why the second part of the equation landed little higher that the first one, sorry for that]
and compare it to the polynomial we get when multyplying [tex]W(z)[/tex] by [tex]\sum_k {1} / (z-z_k)[/tex]. What do you think of it? I've tried to do this, but the calculus grow pretty vast and I feel that there is a simplier method of proving this.

I would appreciate if you could tell me wheter I understand the question right and if my idea of solving it looks fine.

thanks for your time,
rahl.
 
Last edited:
I have found the solution. If no one deletes this thread, I will write how to solve it, maybe it will help someone some day.
 

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