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I was trying to solve the following problem:

Let $f: D \longrightarrow D$ be proper holomorphic (so that means it is a Blaschke product with finitely many factors). Suppose $\{ a_1, ..., a_n \} \subset D$ be the set of branch points of $f$ and the ramification degree of $a_j$ is $m_j \in [2, \infty) \cap \mathbb{N}$ (by this I mean there exists $\alpha_j \in D$ such that $f( \alpha_j) = a_j$ and $\alpha_j$ is a ramification point of degree $m_j$).

The only progress I have made so far is that I can extend this map to be a holomorphic map from the Riemann sphere to itself. Then I can get a lower bound on the degree of the rational map using the Riemann-Hurwitz formula.

I would like to know how to construct such a function and if this problem has connections to any other phenomenon in (algebraic) geometry. Thanks in advance.

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  • $\begingroup$ Also if someone has some resources that could help solve the above, please include them. $\endgroup$
    – nandi
    Mar 29, 2022 at 3:54

1 Answer 1

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This (existence and uniqueness) is proved in the paper in much more general setting (in fact, the result is due to E. Picard):

M. Heins, ‘On a class of conformal metrics’, Nagoya Math. J. 21 (1962) 1–50.

For a more recent and simpler proof of the special case that you ask, see

MR1479037 (99d:30009) Zakeri, Saeed On critical points of proper holomorphic maps on the unit disk. Bull. London Math. Soc. 30 (1998), no. 1, 62–66.

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  • $\begingroup$ Thank you so much!! $\endgroup$
    – nandi
    Mar 29, 2022 at 17:34

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