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Alexandre Eremenko
Alexandre Eremenko
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  1. Besides the position of ramification points $z_j$ you need monodromy of $Q$ to determine $M$. Once $M$ is defined, you need a normalization of your uniformizing function: it is defined up to a conformal automorphism of the sphere $L^1$.

  2. No general formula exists. To find the uniformization in the case of genus $0$, you have to find a rational function with simple critical points and prescribed critical values. The degree of your function is $M+1$ by Riemann-Hurwitz formula. You write it with undetermined coefficients, add normalization, for example $f(0)=a,f(1)=b,f(\infty)=c$, where $a,b,c$ are some points distinct from your $z_j$, and write the condition that critical values are $z_j$. You obtain $2M+3$ algebraic equations with $2M+3$ unknowns, this system has finitely many solutions. Different solutions correspond to different possible monodromies. The system is difficult to solve when the degree is high.

EDIT: You may look at this paper, where a special case of the problem is studied in great detail: Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry, Ann of Math., 155 (2002), 105-129.

  1. Besides the position of ramification points $z_j$ you need monodromy of $Q$ to determine $M$. Once $M$ is defined, you need a normalization of your uniformizing function: it is defined up to a conformal automorphism of the sphere $L^1$.

  2. No general formula exists. To find the uniformization in the case of genus $0$, you have to find a rational function with simple critical points and prescribed critical values. The degree of your function is $M+1$ by Riemann-Hurwitz formula. You write it with undetermined coefficients, add normalization, for example $f(0)=a,f(1)=b,f(\infty)=c$, where $a,b,c$ are some points distinct from your $z_j$, and write the condition that critical values are $z_j$. You obtain $2M+3$ algebraic equations with $2M+3$ unknowns, this system has finitely many solutions. Different solutions correspond to different possible monodromies. The system is difficult to solve when the degree is high.

  1. Besides the position of ramification points $z_j$ you need monodromy of $Q$ to determine $M$. Once $M$ is defined, you need a normalization of your uniformizing function: it is defined up to a conformal automorphism of the sphere $L^1$.

  2. No general formula exists. To find the uniformization in the case of genus $0$, you have to find a rational function with simple critical points and prescribed critical values. The degree of your function is $M+1$ by Riemann-Hurwitz formula. You write it with undetermined coefficients, add normalization, for example $f(0)=a,f(1)=b,f(\infty)=c$, where $a,b,c$ are some points distinct from your $z_j$, and write the condition that critical values are $z_j$. You obtain $2M+3$ algebraic equations with $2M+3$ unknowns, this system has finitely many solutions. Different solutions correspond to different possible monodromies. The system is difficult to solve when the degree is high.

EDIT: You may look at this paper, where a special case of the problem is studied in great detail: Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry, Ann of Math., 155 (2002), 105-129.

Source Link
Alexandre Eremenko
Alexandre Eremenko
  • 91.8k
  • 9
  • 259
  • 429

  1. Besides the position of ramification points $z_j$ you need monodromy of $Q$ to determine $M$. Once $M$ is defined, you need a normalization of your uniformizing function: it is defined up to a conformal automorphism of the sphere $L^1$.

  2. No general formula exists. To find the uniformization in the case of genus $0$, you have to find a rational function with simple critical points and prescribed critical values. The degree of your function is $M+1$ by Riemann-Hurwitz formula. You write it with undetermined coefficients, add normalization, for example $f(0)=a,f(1)=b,f(\infty)=c$, where $a,b,c$ are some points distinct from your $z_j$, and write the condition that critical values are $z_j$. You obtain $2M+3$ algebraic equations with $2M+3$ unknowns, this system has finitely many solutions. Different solutions correspond to different possible monodromies. The system is difficult to solve when the degree is high.