# Limit of critical points of algebraic functions

1. Mar 23, 2012

### jackmell

Hi guys,

I have questions about algebraic functions and not sure where to ask. Hope it's ok here.

Given the algebraic function

$$f(z,w)=a_0(z)+a_1(z)w+\cdots+a_n(z)w^n=0$$

I recall seeing a reference that stated as $n$ increases, the critical points of the function migrate to the unit circle or they migrate to some other circle depending on the orders of $a_i$. Not sure what. However, the trend is nicely suggestive by the four plots below for n=5, 10, 15, 20 where the degree of each $a_i$ is also 5, 10, 15, and 20 respectively and where the set of points is where the number of roots of $f(z_0,w)=0$ is less than $n$. This I compute by setting the resultant of f(z,w) and it's partial with respect to w both equal to zero and that happens only when there is a root of multiplicity greater than one or the point $z_0$ is a solution to $a_n(z)=0$. I'm now unable to find that reference and was hoping someone could help me.

May I ask what exactly do the critical points tend to and how is this proven?

Thanks,
Jack

#### Attached Files:

• ###### critical points of w.jpg
File size:
11.9 KB
Views:
74
Last edited: Mar 23, 2012
2. Mar 24, 2012

### mathwonk

there must be some more hypotheses, since if there is no other condition on an, there is no reason for its roots to migrate anywhere special.

3. Mar 24, 2012

### jackmell

Ok. I'll try back-tracking some more with my references. Still though, it's hard to ignore the trend. Below is a random 30-degree function with each coefficient a_n also 30-degree with coefficients of a_n between -9 and 9. They do seem to be congregating around the unit circle.

#### Attached Files:

• ###### thirtydegreefunction.jpg
File size:
10.9 KB
Views:
66
4. Mar 24, 2012

### micromass

Staff Emeritus
5. Mar 25, 2012

### jackmell

Ok. Thanks a lot. I can see how that would relate to the critical points of an algebraic function since those points are the zeros of the resultant of $f(z,w)$ and $f_w$ and the resultant is a polynomial in $z$.

Last edited: Mar 25, 2012