Geodesic
Imaginary-quadratic solutions of anti-Vandermonde systems in 4~unknowns and the Galois orbits of trees of diameter~4
Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 3, pp. 229-236.

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

The paper is devoted to an elementary Diophantine problem motivated by Grothendieck's dessins d'enfants theory. Namely, we consider the system of equations $ax^j+by^j+cz^j+dt^j=0$ ($j=1,2,3$) with natural $a$, $b$, $c$, and $d$. For trivial reasons it has no real (hence rational) nonzero solutions; we study the cases where it has imaginary quadratic ones. We suggest an infinite family of such cases covering all the imaginary quadratic fields. We discuss this result from the viewpoint of the Galois orbits of trees of diameter 4. 
@article{FPM_2003_9_3_a15,
     author = {G. B. Shabat},
     title = {Imaginary-quadratic solutions of {anti-Vandermonde} systems in 4~unknowns and the {Galois} orbits of trees of diameter~4},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {229--236},
     publisher = {mathdoc},
     volume = {9},
     number = {3},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2003_9_3_a15/}
}
TY  - JOUR
AU  - G. B. Shabat
TI  - Imaginary-quadratic solutions of anti-Vandermonde systems in 4~unknowns and the Galois orbits of trees of diameter~4
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2003
SP  - 229
EP  - 236
VL  - 9
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2003_9_3_a15/
LA  - ru
ID  - FPM_2003_9_3_a15
ER  - 
%0 Journal Article
%A G. B. Shabat
%T Imaginary-quadratic solutions of anti-Vandermonde systems in 4~unknowns and the Galois orbits of trees of diameter~4
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2003
%P 229-236
%V 9
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2003_9_3_a15/
%G ru
%F FPM_2003_9_3_a15
G. B. Shabat. Imaginary-quadratic solutions of anti-Vandermonde systems in 4~unknowns and the Galois orbits of trees of diameter~4. Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 3, pp. 229-236. http://geodesic.mathdoc.fr/item/FPM_2003_9_3_a15/

[1] Adrianov N. M., Kochetkov Yu. Yu., Suvorov A. D., Shabat G. B., “Gruppy Mate i ploskie derevya”, Fundam. i prikl. mat., 1:2 (1995), 377–384 | MR | Zbl

[2] Kochetkov Yu. Yu., Chastnoe soobschenie, 1995

[3] Kochetkov Yu. Yu, Chastnoe soobschenie, 1996

[4] Grothendieck A., “Esquisse d'un programme”, Geometric Galois Actions, London Math. Society, Lecture Notes Series, 243, Cambridge Univ. Press, 1977, 3–43

[5] Kochetkov Yu., “Trees of diameter $4$”, Formal Power Series and Algebraic Combinatorics, Proceedings of the 12th International Conference, FPSAC'00, eds. D. Krob, A. A. Mikhalev, A. V. Mikhalev, Springer, 2000, 447–475 | MR

[6] Schneps L., “Dessins d'enfants on the Riemann sphere”, The Grothendieck Theory of Dessins d'Enfants, London Math. Society, Lecture Notes Series, 200, ed. Schneps L., Cambridge Univ. Press, 1944, 47–98 | MR

[7] Shabat G., Voevodsky V., “Drawing curves over number fields”, The Grothendieck Festschrift, Vol. 3, Birkhäuser, 1990, 199–227 | MR | Zbl

[8] Shabat G., Zvonkin A., “Plane trees and algebraic numbers”, Jerusalem Combinatorics '93, Contemporary Mathematics, 178, Amer. Math. Soc., Providence, RI, 1994, 233–275 | MR | Zbl

[9] Zapponi L., “Fleurs, arbres et cellules: un invariant galoisien pour une famille d'arbres”, Compositio Mathematika, 122 (2000), 113–133 | DOI | MR | Zbl

[10] Schneps L. (ed.), The Grothendieck Theory of Dessins d'Enfants, London Math. Society, Lecture Notes Series, 200, Cambridge Univ. Press, 1944