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Minimality of the D-groupoid of symmetries of a projective structure
Minimalidad del D-grupoide de simetrías de una estructura proyectiva
DOI:
https://doi.org/10.15446/recolma.v58n2.121031Palabras clave:
Schwarzian equation, Schwarzian derivative, Strong minimality, Symmetric Power, Lie groupoid, D-groupoid (en)derivada Schwarziana, fuerte minimalidad, grupoide de Lie, D-grupoide, Ecuación Schwarziana, potencia simétrica (es)
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In this article we study Kummer’s D-groupoid, which is the groupoid of symmetries of a meromorphic projective structure. We give necessary and sufficient conditions for its minimality, in the sense of not having infinite sub-D-groupoids. The condition that we find turns out to be equivalent to the strong minimality of the non-linear Schwarzian equation and the non-integrability by means of Liouvillian functions of the linear Schwarzian equation.
En este artículo estudiamos el D-grupoide de Kummer, el cual es el grupoide de simetrías de una estructura proyectiva meromorfa. Damos condiciones necesarias y suficientes para su simplicidad, en el sentido de no tener sub-D-grupoides no finitos. La condición que encontramos resulta ser equivalente a la fuerte minimalidad de la ecuación schwarziana no lineal y la no integrabilidad mediante funciones liouvillianas de la ecuación schwarziana lineal.
Referencias
[1] V. Aslanyan, Ax-Schanuel and strong minimality for the j-function, Annals of Pure and Applied Logic 172 (2021), no. 1, 102871. DOI: https://doi.org/10.1016/j.apal.2020.102871
[2] D. Bl´azquez-Sanz, G. Casale, and J. S. Arboleda, The Malgrange-Galois groupoid of the Painlev´e VI equation with parameters, Advances in Geometry 22 (2022), no. 3, 301-328. DOI: https://doi.org/10.1515/advgeom-2022-0010
[3] D. Bl´azquez-Sanz, G. Casale, J. Freitag, and J. Nagloo, Some functional transcendence results around the Schwarzian differential equation., Annales de la Facult´e des sciences de Toulouse: Math´ematiques, vol. 29, 2020, pp. 1265-1300. DOI: https://doi.org/10.5802/afst.1661
[4] G. Casale, Enveloppe galoisienne d’une application rationnelle de P1, Publ. Mat. 50 (2006), no. 1, 191-202. MR 2325017 DOI: https://doi.org/10.5565/PUBLMAT_50106_10
[5] G. Casale, J. Freitag, and J. Nagloo, Ax-Lindemann-Weierstrass with derivatives and the genus 0 fuchsian groups, Annals of Mathematics 192 (2020), no. 3, 721-765. DOI: https://doi.org/10.4007/annals.2020.192.3.2
[6] T. Crespo and Z. Hajto, Algebraic groups and differential Galois theory, Graduate studies in mathematics, vol. 122, American Mathematical Soc., 2011. DOI: https://doi.org/10.1090/gsm/122
[7] H. P. De Saint-Gervais, Uniformization of Riemann surfaces: Revisiting a hundred-year-old theorem, Heritage of European Mathematics, EMS, European Mathematical Society, 2016. DOI: https://doi.org/10.4171/145
[8] J. ´Ecalle, Les fonctions r´esurgentes. Tome II, Publications Math´ematiques d’Orsay 81 [Mathematical Publications of Orsay 81], vol. 6, Universit´e de Paris-Sud, D´epartement de Math´ematiques, Orsay, 1981, Les fonctions r´esurgentes appliqu´ees `a l’it´eration. [Resurgent functions applied to iteration]. MR 670418
[9] J. Harnad, Picard-fuchs equations, hauptmoduls and integrable systems. integrability: the seiberg-witten and whitham equations (edinburgh, 1998), 2000.
[10] B. Klingler, E. Ullmo, and A. Yafaev, The hyperbolic Ax-Lindemann- Weierstraß conjecture, Publications math´ematiques de l’IH´ES 123 (2016), no. 1, 333-360. DOI: https://doi.org/10.1007/s10240-015-0078-9
[11] J. J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, Journal of Symbolic Computation 2 (1986), no. 1, 3-43. DOI: https://doi.org/10.1016/S0747-7171(86)80010-4
[12] E. E. Kummer, ¨uber die hypergeometrische Reihe ., 1836 (1836), no. 15, 39-83. DOI: https://doi.org/10.1515/crll.1836.15.39
[13] K. Mahler, On algebraic differential equations satisfied by automorphic functions, Doc. Math. (2019), 661-667, Reprint of [0262493]. MR 4605031 DOI: https://doi.org/10.4171/dms/8/41
[14] B. Malgrange, Le groupo¨ıde de Galois d’un feuilletage, L’enseignement math´ematique 38 (2001), no. 2, 465-501.
[15] B. Malgrange, Pseudogroupes de Lie et th´eorie de Galois diff´erentielle, IHES, 2010.
[16] J. Pila, Modular Ax-Lindemann-Weierstrass with derivatives, Notre Dame Journal of Formal Logic 54 (2013), no. 3-4. DOI: https://doi.org/10.1215/00294527-2143853
[17] J. Pila and J. Tsimerman, Ax-Schanuel for the j-function, Duke Mathematical Journal 165 (2016), no. 13, 2587-2605. DOI: https://doi.org/10.1215/00127094-3620005
[18] J. F. Ritt, Transcendental transcendency of certain functions of Poincar´e, Math. Ann. 95 (1926), no. 1, 671–682. MR 1512299 DOI: https://doi.org/10.1007/BF01206632
[19] M. Van der Put and M. F. Singer, Galois theory of linear differential equations, Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, 2003. DOI: https://doi.org/10.1007/978-3-642-55750-7
[20] E. W. Weisstein, Faa di bruno’s formula, https://mathworld. wolfram.com/ (2003).
[21] M. Yoshida, Fuchsian differential equations: With special emphasis on the Gauss-Schwarz theory, Aspects of Mathematics, Vieweg+Teubner Verlag, 2013.
