Parsing the Fibers of Critical Values

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WWGD

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Hi, All:

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

Specifically, I had the fibers of the map:

f(z1,z2): ℂ2→ℂ , given by:

f(Z1, Z2)= Z12+z22 =0

I rewrote it as a map from ℝ4→ℝ2=

f(x1,y1,x2,x2)=

(x12-y12+x22-

y22 , x1y1+2x2y2)


Which has the Jacobian:
(Sorry, I don't know how to Tex a matrix ):

[ 2x1, 2y1, 2x2, 2y2 ]

[ 2y1, 2x1, 2y2, 2x2





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 ℂ2.

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: Z12+Z22=0

implies Z1=iZ2.

ii) A little better:

Restricting to (x1,x2, 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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