4 misprint
1. There is a characterization of Schwarzian derivatives of rational maps: section 3 in the text: http://www.math.purdue.edu/~eremenko/Pdf/schwarz.pdfhttp://www.math.purdue.edu/~eremenko/dvi/schwarz.pdf There is something similar also in arXiv:math/0512370, chapter 2. All these descriptions are various systems of algebraic equations. One of them, the "Bethe ansatz equations for the Gaudin model", proved to be very useful, see Mukhin, Tarasov and Varchenko, The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Ann. Math. 170, 2 2009, 863-15.

2. There is some cell decomposition of the sphere which can be intrinsically related to a ratonal function. It is described in the paper Bonk, Eremenko, Schlicht regions of entire and meromorphic functions, J. d'Analyse, 77, 1999, 69-104, Sections 7.8. For a given cell decomposition, a rational function can be recovered using an algorithm similar to Thurston's circle packing algorithm. However, with this description, critical points or critical valued cannot be prescribed, and the cell decomposition does not determine the rational function completely.

Alex Eremenko.

3 misprint corrected
1. There is a characterization of Schwarzian derivatives of rational maps: section 3 in the text: http://www.math.purdue.edu/~eremenko/Pdf/schwarz.pdf There is something similar also in arXiv:math/0512370, chapter 2. All these descriptions are various systems of algebraic equations. One of them, the "Bethe ansatz equations for the Gaudin model", proved to be very useful, see Mukhin, Tarasov and Varchenko, The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Ann. Math. 170, 2 2009, 863-15.

2. There is some cell decomposition of the sphere which can be intrinsically related to a ratonal function. It is described in the paper Bonk, Eremenko, Schlicht regions of entire and meromorphic functions, J. d'Analyse, 77, 1999, 69-104, Sections 7.8. For a given cell decomposition, the a rational function can be recovered using an algorithm similar to Thurston's circle packing algorithm. However, with this description, critical points or critical valued cannot be prescribed, and the cell decomosition decomposition does not determine the rational function completely.

Alex Eremenko.

2 misprint corrected
1. There is a characterization of Schwarzian derivatives of rational maps: section 3 in the text: http://www.math.purdue.edu/~eremenko/Pdf/schwarz.pdf There is something similar also in arXiv:math/0512370, chapter 2. All these descriptions are various systems of algebraic equations. One of them, the "Bethe ansatz equations for the Gaudin model", proved to be very useful, see Mukhin, Tarasov and Varchenko, The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Ann. Math. 170, 2 2009, 863-15.

2. There is some cell decomposition of the sphere which can be intrinsically related to a ratonal function. It is described in the paper Bonk, Eremenko, Schlicht regions of entire and meromorphic functions, J. d'Analyse, 77, 1999, 69-104, Sections 7.8. For a given cell decomposition, the rational function can be recovered using an algorithm similar to Thurston's circle packing algorithm. However, with this description, critical points or critical valued cannot be prescribed: , and the cell decomosition itself already determines does not determine the rational functioncompletely.

Alex Eremenko.

1