# zbMATH — the first resource for mathematics

An ultrametric version of the Maillet-Malgrange theorem for nonlinear $$q$$-difference equations. (English) Zbl 1152.33011
An analog of the Maillet-Malgrange theorem for ultrametric nonlinear $$q$$-difference equation is established in this paper, under the assumption $$\mod (q)$$ not equal to unity. The result so obtained, generalizes to nonlinear $$q$$-difference equation, a theorem of J.-P. Bézivin and A. Boutabaa [Collect. Math. 43, No. 2, 125–140 (1992; Zbl 0778.39009)]. In Section 2, examples of the result are given, Section 3 gives the proof of the main result. The author claims to have obtained a much stronger version of Theorem 7 (page 2811, this paper). The references are exhaustive and some of them may encourage researchers to study Gevrey theory for linear $$q$$-difference equations.

##### MSC:
 33E99 Other special functions 39A13 Difference equations, scaling ($$q$$-differences)
Full Text:
##### References:
 [1] Norbert A’Campo, Théorème de préparation différentiable ultra-métrique, Séminaire Delange-Pisot-Poitou: 1967/68, Théorie des Nombres, Fasc. 2, Exp. 17, Secrétariat mathématique, Paris, 1969, pp. 7. · Zbl 0214.37904 [2] Jean-Paul Bézivin and Abdelbaki Boutabaa, Sur les équations fonctionelles \?-adiques aux \?-différences, Collect. Math. 43 (1992), no. 2, 125 – 140 (French, with English summary). · Zbl 0778.39009 [3] Jean-Paul Bézivin, Convergence des solutions formelles de certaines équations fonctionnelles, Aequationes Math. 44 (1992), no. 1, 84 – 99 (French, with English summary). · Zbl 0762.39012 [4] Jean-Paul Bézivin, Sur les équations fonctionnelles aux \?-différences, Aequationes Math. 43 (1992), no. 2-3, 159 – 176 (French, with English summary). · Zbl 0757.39002 [5] D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen, Plane curves associated to character varieties of 3-manifolds, Invent. Math. 118 (1994), no. 1, 47 – 84. · Zbl 0842.57013 [6] Lucia Di Vizio, Arithmetic theory of \?-difference equations: the \?-analogue of Grothendieck-Katz’s conjecture on \?-curvatures, Invent. Math. 150 (2002), no. 3, 517 – 578. · Zbl 1023.12004 [7] Lucia Di Vizio. Local analytic classification of $$q$$-difference equations with $$| q|=1$$. arXiv:0802.4223 · Zbl 1168.39002 [8] L. Di Vizio, J.-P. Ramis, J. Sauloy, and C. Zhang, Équations aux \?-différences, Gaz. Math. 96 (2003), 20 – 49 (French). · Zbl 1063.39015 [9] Monique Fleinert-Jensen, Théorèmes d’indices précisés et convergences des solutions pour une équation linéaire aux \?-différences, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 4, 425 – 428 (French, with English and French summaries). · Zbl 0898.39008 [10] Stavros Garoufalidis, On the characteristic and deformation varieties of a knot, Proceedings of the Casson Fest, Geom. Topol. Monogr., vol. 7, Geom. Topol. Publ., Coventry, 2004, pp. 291 – 309. · Zbl 1080.57014 [11] K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, and Y. Yamada, Construction of hypergeometric solutions to the \?-Painlevé equations, Int. Math. Res. Not. 24 (2005), 1441 – 1463. · Zbl 1082.33013 [12] Edmond Maillet. Sur les séries divergentes et les équations différentielles. Annales Scientifiques de l’École Normale Supérieure. Troisième Série, 20, 1903. [13] Bernard Malgrange, Sur le théorème de Maillet, Asymptotic Anal. 2 (1989), no. 1, 1 – 4 (French). · Zbl 0693.34004 [14] Fabienne Naegele, Théorèmes d’indices pour les équations \?-différences-différentielles, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 6, 579 – 582 (French, with English and French summaries). · Zbl 0779.39001 [15] J.-P. Ramis, Dévissage Gevrey, Journées Singulières de Dijon (Univ. Dijon, Dijon, 1978) Astérisque, vol. 59, Soc. Math. France, Paris, 1978, pp. 4, 173 – 204 (French, with English summary). [16] A. Ramani, B. Grammaticos, T. Tamizhmani, and K. M. Tamizhmani, Special function solutions of the discrete Painlevé equations, Comput. Math. Appl. 42 (2001), no. 3-5, 603 – 614. Advances in difference equations, III. · Zbl 0994.33500 [17] Jean-Pierre Serre, Lie algebras and Lie groups, Lecture Notes in Mathematics, vol. 1500, Springer-Verlag, Berlin, 2006. 1964 lectures given at Harvard University; Corrected fifth printing of the second (1992) edition. [18] Yasutaka Sibuya and Steven Sperber, Convergence of power series solutions of \?-adic nonlinear differential equation, Recent advances in differential equations (Trieste, 1978) Academic Press, New York-London, 1981, pp. 405 – 419. · Zbl 0602.12009 [19] Yasutaka Sibuya and Steven Sperber, Some new results on power-series solutions of algebraic differential equations, Singular perturbations and asymptotics (Proc. Adv. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1980) Publ. Math. Res. Center Univ. Wisconsin, vol. 45, Academic Press, New York-London, 1980, pp. 379 – 404. · Zbl 0515.34013 [20] Yasutaka Sibuya and Steven Sperber, Arithmetic properties of power series solutions of algebraic differential equations, Ann. of Math. (2) 113 (1981), no. 1, 111 – 157. · Zbl 0492.12012 [21] Changgui Zhang, Sur un théorème du type de Maillet-Malgrange pour les équations \?-différences-différentielles, Asymptot. Anal. 17 (1998), no. 4, 309 – 314 (French, with English and French summaries). · Zbl 0938.34064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.