Abstract
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.
Similar content being viewed by others
References
N. M. Adrianov, Yu. Yu. Kochetkov, A. D. Suvorov, and G. B. Shabat, “Mathieu groups and plane trees,” Fund. Prikl. Mat., 1995, 1, No. 2, 377–384 (1995).
A. Grothendieck, “Esquisse d’un programme,” in: Geometric Galois Actions, Cambridge Univ. Press (1977), London Math. Society, Lecture Notes Series, Vol. 243, pp. 3–43.
Yu. Kochetkov, private communication (1995).
Yu. Kochetkov, private communication (1996).
Yu. Kochetkov, “Trees of diameter 4,” in: D. Krob, A. A. Mikhalev, and A. V. Mikhalev, eds., Proceedings of the 12th International Conference, FPSAC’00 (Formal Power Series and Algebraic Combinatorics), Springer (2000), pp. 447–475.
L. Schneps, “Dessins d’enfants on the Riemann sphere,” in: L. Schneps, ed., The Grothendieck Theory of Dessins d’Enfants, Cambridge Univ. Press (1944), London Math. Society, Lecture Notes Series, Vol. 200, pp. 47–98.
L. Schneps, ed., The Grothendieck Theory of Dessins d’Enfants, Cambridge Univ. Press (1944), London Math. Society, Lecture Notes Series, Vol. 200.
G. Shabat and V. Voevodsky, “Drawing curves over number fields,” in: The Grothendieck Festschrift, Vol. 3, Birkhäuser (1990), pp. 199–227.
G. Shabat and A. Zvonkin, “Plane trees and algebraic numbers,” in: Jerusalem Combinatorics ’93, Contemporary Mathematics, Amer. Math. Soc., Vol. 178, 1994, pp. 233–275.
L. Zapponi, “Fleurs, arbres et cellules: un invariant galoisien pour une famille d’arbres,” Compositio Math., 122, 113–133 (2000).
Author information
Authors and Affiliations
Additional information
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 229–236, 2003.
Rights and permissions
About this article
Cite this article
Shabat, G.B. Imaginary-quadratic solutions of anti-Vandermonde systems in 4 unknowns and the Galois orbits of trees of diameter 4. J Math Sci 135, 3420–3424 (2006). https://doi.org/10.1007/s10958-006-0171-1
Issue Date:
DOI: https://doi.org/10.1007/s10958-006-0171-1