- #1

jackmell

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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

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

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