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Can Mathematica reduce this inequality into a domain of z, where {z, r1, r2} are all complex?

Reduce[Abs[z^2 - r1] > Abs[z^3 - r2], z]

The line never returns a result. I also tried just calculating the boundary:

Simplify[Abs[z^2 - r1] == Abs[z^3 - r2]]

But that never returns a result either. I've tried a few other methods that all just crashed or failed. I found a related Question and another related Question but I couldn't figure out how to apply those partial solutions to this problem.

BTW, I realize Solve[] and Reduce[] don't cope with Abs[]. Even this simpler command give a poor result:

Simplify[Abs[z^2] == Abs[z^3]];

(* Abs[z]^2 == Abs[z] *)

And just to be clear, I don't just need the answer to the inequality, I need to know how to make Mathematica do it.

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Jerry Guern
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1 Answers1

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You need to specify all of the variables you want to reduce over: 

In[1]:= Reduce[Abs[z^2 - r1] > Abs[z^3 - r2], {z, r1, r2}, Complexes]
Out[1]= (* a huge pile of Im, Re and Sqrt omitted *)
Pillsy
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  • Thanks for the tip about adding Complexes, but yikes that huge pile result is not a good result. I don't think it should be necessary to explicitly separate Im[z] and Re[z] in the result. – Jerry Guern Nov 13 '15 at 23:08