Geodesic
Fields of definition of rational functions of one variable with three critical values
Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 3, pp. 781-799 
V. O. Filimonenkov; G. B. Shabat. Fields of definition of rational functions of one variable with three critical values. Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 3, pp. 781-799. http://geodesic.mathdoc.fr/item/FPM_1995_1_3_a13/
@article{FPM_1995_1_3_a13,
     author = {V. O. Filimonenkov and G. B. Shabat},
     title = {Fields of definition of rational functions of one variable with three critical values},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {781--799},
     year = {1995},
     volume = {1},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1995_1_3_a13/}
}
TY  - JOUR
AU  - V. O. Filimonenkov
AU  - G. B. Shabat
TI  - Fields of definition of rational functions of one variable with three critical values
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 1995
SP  - 781
EP  - 799
VL  - 1
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/FPM_1995_1_3_a13/
LA  - ru
ID  - FPM_1995_1_3_a13
ER  - 
%0 Journal Article
%A V. O. Filimonenkov
%A G. B. Shabat
%T Fields of definition of rational functions of one variable with three critical values
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 1995
%P 781-799
%V 1
%N 3
%U http://geodesic.mathdoc.fr/item/FPM_1995_1_3_a13/
%G ru
%F FPM_1995_1_3_a13

Voir la notice de l'article provenant de la source Math-Net.Ru

Rational functions of one complex variable with three critical values are considered. According to Grothendieck theory, such functions are classified by the isotopic classes of the connected spherical graphs. By a suitable fractional-linear variable substitution these coefficients can be transformed to algebraic numbers. The paper is devoted to the problem of minimization of irrationality degrees of these coefficients.

[1] Belyi G. B., “O rasshirenii Galua maksimalnogo krugovogo polya”, Izvestiya AN SSSR, ser. matem., 43:2 (1979), 267–276 | MR | Zbl

[2] Shafarevich I. R., “Osnovnye ponyatiya algebry”, Algebra-1, Itogi nauki i tekhniki. Seriya sovremennye problemy matematiki. Fundamentalnye napravleniya, 11, VINITI, Moskva, 1986 | MR

[3] Serr Zh.-P., Kogomologii Galua, Mir, M., 1968 | MR

[4] Filimonenkov V. O., “Ob arifmetike ratsionalnykh funktsii pri drobnolineinykh zamenakh argumenta”, Vestnik MGU, ser. matem., mekhan., 1995, no. 1, 90–92 | MR

[5] Couveighes J.-M., “Calcul et rationalite de fonction de Belyi en genre 0”, Ann. de l'inst. Fourier, 44 (1) (1994) | MR

[6] Grothendieck A., Esquisse d'un programme, Preprint, 1984 | MR

[7] Shabat G., Voevodsky V., “Drawing curves over number fields”, 111, The Grothendiek Festschrift, Birkhauser, 1990, 199–227 | MR | Zbl