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We need to Find a non constant map $f:\mathbb{C}^3\to \mathbb{C}$ such that for any three distinct complex numbers $z_1,z_2,z_3$ and any automorphism $\phi$ of $\mathbb{C}$, we have $f(z_1,z_2,z_3)= f(\phi(z_1),\phi(z_2),\phi(z_3))$

Thank you for help and discussion.

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Whoops, I understand now that the analytic sense of automorphism (holomorphic bijection) is meant. – Sam Hopkins Mar 9 '13 at 6:30
up vote 1 down vote accepted

If the $z_i$s are all distinct, take

$f(z_1,z_2,z_3) = \frac{z_1-z_2}{z_3-z_2}$

This is the cross-ratio. It can take any complex value except for $0$ and $1$. It is clearly invariant. You can get many other functions by composing this with some function from $\mathbb C - \{0,1\}$ to $\mathbb C$. Indeed, these functions are all the possible functions, since if two triples have the same cross-ratio, they are the same up to an automorphism.

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