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I am trying to better understand what happens at fibers of critical values:

Specifically, I had the fibers of the map:

f(z1,z2): ℂ^{2}→ℂ , given by:

f(Z_{1}, Z_{2})= Z_{1}^{2}+z_{2}^{2}=0

I rewrote it as a map from ℝ^{4}→ℝ^{2}=

f(x_{1},y_{1},x_{2},x_{2})=

(x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-

y_{2}^{2}, x_{1}y_{1}+2x_{2}y_{2})

Which has the Jacobian:

(Sorry, I don't know how to Tex a matrix ):

[ 2x_{1}, 2y_{1}, 2x_{2}, 2y_{2}]

[ 2y_{1}, 2x_{1}, 2y_{2}, 2x_{2}

Now, by one of the family of theorems in { inverse function, implicit function, regular rank theorem} , the fibers of the non-zero values are submanifolds of ℂ^{2}.

After row-reducing, we get that the only critical value is (0,0) .

Now, some work on f^{{-1}}(0,0):

i)Not very helpful: Z_{1}^{2}+Z_{2}^{2}=0

implies Z_{1}=iZ_{2}.

ii) A little better:

Restricting to (x_{1},x_{2}, 0, 0):

Then the general fiber is a circle, and the collection of all fibers is a cone .

Is there anything else we can do to get a better understanding of what the general

fibers are like ? I think all the regular fibers are isomorphic to each other--tho I cannot

find a good proof -- and then we have the critical fiber.

Any ideas?

Thanks.

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# Parsing the Fibers of Critical Values

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