Ch 5, Exercise 10

Gary

This is my second reply. Please check first reply above.
I have looked at your solution.  Here are some comments:

1. I think that when writing the derivative $dP(z)/dz$ using a dash, it should by convention be $P'(z)$. Writing $P(z)'$ could be misinterpreted as $P\cdot (z)'$

2. On page 2 there is a sentence near the bottom about the hint, which I don't think is relevant here.  If you just have points then you assume that the mass at each point is the same.  The hint is about the result 32 on page 104 being true when the masses are not all equal.

3. I think that the paragraph in the middle of your page 4 could be amended to say "...the equation in 5;" rather than (i)., and where you write "...centroid and critical point...", since these are the same thing, you could also rewrite this to avoid confusion.

4.  In the same paragraph you write "There will be a different critical point for each set of $a_j$, that is for each $P(z)$". Although I agree that if we change the $a_j$ we are dealing with a different $P$ and therefore different critical points, in the proof here we are dealing with a specific $P$ with zeros $a_j$ and a corresponding set of critical points, and each critical point is the centroid of a different set of masses at the zeros of $P$.  Perhaps it would be better to write: " There will be a different set of masses at the zeros $a_j$, for each critical point.

5. In your discussion of figure 1, which starts at the bottom of page 4 and continues onto page 5, you say at the top of page 5: "If the answers...the centre of mass of $\mathcal{I}_K(a_j)$." I don't think that this is correct for reasons I have detailed in my first reply and also because the centre of mass of $\mathcal{I}_K(a_j)$ must coincide with the centre of $K$, the circle which has produced the $\mathcal{I}_K(a_j)$, for what you say to be true. This is difficult to explain - I hope it makes sense.

Vasco

Gary

Here are my responses to your last post.



The equation in (iii) is equivalent to $P'(z)=0$. By rewriting it in the form $z=\frac{\sum m_ja_j}{\sum m_j}$ it is still equivalent to $P'(z)=0$. It is not an equation which allows you to calculate $z$. It's a bit like saying that if $P'(z)=6z^2-2=0$ then we can rewrite it as $z=1/(3z)$: the values of $z$ which satisfy the equation $6z^2-2=0$ and the values of $z$ which satisfy $z=1/(3z)$ are the same. We have not solved for $z$ however, just stated things in a different way. However, any $z$ which satisfies either equation is a solution i.e. a critical point of $P(z)$ or, equivalently, a zero of $P'(z)$.



In part (i) we show that $P'(z)=0$ can be re-written as $1/(z-a_1)+..=0$ so that the critical points are the $n-1$ values of $z$ which satisfy this equation. In (ii) we show that if we draw a circle $K$ centred at $p$, then $z=p$ will be a critical point of $P$ if and only if we arrange things so that $p$ is the centre of mass of the inverted points $\mathcal{I}_K(a_j)$.
In part (iii) we are NOT saying that $z = \frac{\sum m_j a_j}{\sum m_j}$ is a critical point, we are saying that any $z$ which satisfies this equation is a critical point of $P$. Given ANY value for $z$, say $q$, we could calculate the values of $m_j$ and then calculate the value of $\frac{\sum m_j a_j}{\sum m_j}$, but in general this would not be equal to the original value of $q$ of $z$. Only if our original choice of $q$ was a critical point of $P$ would we find that $q=\frac{\sum m_j(q)a_j}{\sum m_j(q)}$. I have written $m_j(q)$ as a reminder that $m_j$ is a function of $q$.
The expression $z = \frac{\sum m_j a_j}{\sum m_j}$ is NOT a formula for calculating the critical points, but rather an equation which is satisfied if $z$ is a critical point.

Note1: Part (ii) of this exercise does not seem to contribute to the result in part (iii) - it just seems to be an interesting incidental result.

Note2: You may already know this, but solving for the roots of a general polynomial in $z$ is a non-trivial task.

Vasco

Gary
I'm glad you find it helpful.

I understand it as "...the centroids of the masses $m_j=1/|z-a_j|$ located at the zeros of $P$, the $a_j$" and also I think you should say "the solutions to $P'=0$".

Vasco