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Limit of critical points of algebraic functions

  1. Mar 23, 2012 #1
    Hi guys,

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

    Given the algebraic function

    [tex]f(z,w)=a_0(z)+a_1(z)w+\cdots+a_n(z)w^n=0[/tex]

    I recall seeing a reference that stated as [itex]n[/itex] increases, the critical points of the function migrate to the unit circle or they migrate to some other circle depending on the orders of [itex]a_i[/itex]. 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 [itex]a_i[/itex] is also 5, 10, 15, and 20 respectively and where the set of points is where the number of roots of [itex]f(z_0,w)=0[/itex] is less than [itex]n[/itex]. 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 [itex]z_0[/itex] is a solution to [itex]a_n(z)=0[/itex]. 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:

    Last edited: Mar 23, 2012
  2. jcsd
  3. Mar 24, 2012 #2

    mathwonk

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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.
     
  4. Mar 24, 2012 #3
    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:

  5. Mar 24, 2012 #4

    micromass

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  6. Mar 25, 2012 #5
    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 [itex]f(z,w)[/itex] and [itex]f_w[/itex] and the resultant is a polynomial in [itex]z[/itex].
     
    Last edited: Mar 25, 2012
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