[complex functions] polynomial roots in a convex hull

[rahl___]

Hi everyone,

I've got this problem to solve:

Let $$W(z)$$ be a polynomial with complex coefficients and complex roots. Show that the roots of $$W'(z)$$ are in a convex hull of the set of roots of $$W(z)$$.

My problem is that I don't fully understand the question.

I have found such definition of convex hull:

Given a set of points $$(z_1, z_2, ..., z_n)$$, we denote convex hull as:
$$conv(z_1, z_2, ..., z_n) = \{ z = \sum_{k=1}^n \beta_k z_k : \beta_k \in [0,1], \sum_{k=1}^n \beta_k = 1 \}$$

So I do have to prove, that all the roots of $$W'(z)$$ [let's denote them as $$z'_k$$] must be able to be written in such form:
$$z'_k = \sum_{k=1}^n \beta_k z_k$$, where $$\beta_k$$ are satysfying the conditionsof convex hull and $$z_k$$ are the roots of $$W(z)$$.
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 $$W'(z)$$ as:
$$z'_j = \sum_{k=1}^n \beta_k^{(j)} z_k$$
Now we write down $$W'(z)$$ using viete's formulas:
$$W'(z) = n a_n z^{n-1} - n a_n ( \sum_j \sum_k \beta_k^{(j)} z_k ) z^_{n-2} + n a_n \sum_{i<j} ( \sum_k \beta_k^{(i)} z_k ) ( \sum_k \beta_k^{(j)} z_k ) z^{n-3} - ... + n a_n \prod_j \sum_k \beta_k^{(j)} z_k$$ [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 $$W(z)$$ by $$\sum_k {1} / (z-z_k)$$. 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.

 

[rahl___]

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.

 

[Architeuthis Dux]

Hi rahl,

Your understanding of the question is correct. The goal is to show that the roots of W'(z) can be written in the convex hull form using the roots of W(z). Your approach of using Vieta's formulas is a good start, but as you mentioned, it can become quite complex and difficult to solve.

Another method to approach this problem is to use the fact that the derivative of a polynomial at a root is equal to 0. This means that for each root z_k of W(z), the derivative W'(z) will have a factor (z-z_k). This can be used to write W'(z) as a product of linear factors (z-z_k), which can then be rearranged to fit the convex hull form.

I hope this helps and good luck with solving the problem!