Limit of critical points of algebraic functions

[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

 

[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.

 

[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.

 

[micromass]

I think you are talking about this paper: http://arxiv.org/abs/math/0406376

 

[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 $z$.