# Limit of critical points of algebraic functions

• jackmell
In summary, the conversation is about the behavior of critical points of algebraic functions as the degree of the function increases. The trend is that they migrate towards the unit circle or some other circle depending on the orders of the coefficients. This has been observed in plots for different degrees and orders of coefficients. The person is looking for a reference that explains this trend and how it is proven. They mention a paper that might be relevant and continue to discuss the behavior of critical points.
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

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• critical points of w.jpg
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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.

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.

#### Attachments

• thirtydegreefunction.jpg
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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 $z$.

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## 1. What is a critical point of an algebraic function?

A critical point of an algebraic function is a point on the graph where the derivative of the function is equal to zero or undefined. This means that the slope of the function at that point is either flat or vertical.

## 2. How do you find the critical points of an algebraic function?

To find the critical points of an algebraic function, you need to take the derivative of the function and set it equal to zero. Then, solve for the x-values that make the derivative equal to zero. These x-values are the critical points.

## 3. Why are critical points important in algebraic functions?

Critical points are important in algebraic functions because they give us information about the shape of the graph. They can indicate where the function has maximum or minimum values, or where the graph changes from increasing to decreasing or vice versa.

## 4. Can an algebraic function have more than one critical point?

Yes, an algebraic function can have more than one critical point. In fact, most functions have multiple critical points. This is because the derivative can be equal to zero at multiple x-values.

## 5. How can the limit of critical points be used to analyze an algebraic function?

The limit of critical points can be used to analyze an algebraic function by giving us information about the behavior of the function at these points. By taking the limit as x approaches a critical point, we can determine if the critical point is a local maximum or minimum, or if the function is continuous or discontinuous at that point.

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