An ultrametric version of the Maillet-Malgrange theorem for nonlinear 𝑞-difference equations

Di Vizio, Lucia

doi:10.1090/S0002-9939-08-09352-0

An ultrametric version of the Maillet-Malgrange theorem for nonlinear $q$-difference equations
HTML articles powered by AMS MathViewer

by Lucia Di Vizio

Proc. Amer. Math. Soc. 136 (2008), 2803-2814

DOI: https://doi.org/10.1090/S0002-9939-08-09352-0

Published electronically: March 21, 2008

PDF | Request permission

Abstract:

We prove an ultrametric $q$-difference version of the Maillet- Malgrange theorem on the Gevrey nature of formal solutions of nonlinear analytic $q$-difference equations. Since $\deg _q$ and ${ord}_q$ define two valuations on $\mathbb {C}(q)$, we obtain, in particular, a result on the growth of the degree in $q$ and the order at $q$ of formal solutions of nonlinear $q$-difference equations, when $q$ is a parameter. We illustrate the main theorem by considering two examples: a $q$-deformation of “Painlevé II”, for the nonlinear situation, and a $q$-difference equation satisfied by the colored Jones polynomials of the figure $8$ knots, in the linear case.

We also consider a $q$-analog of the Maillet-Malgrange theorem, both in the complex and in the ultrametric setting, under the assumption that $|q|=1$ and a classical diophantine condition.

References

Norbert A’Campo, Théorème de préparation différentiable ultra-métrique, Séminaire Delange-Pisot-Poitou: 1967/68, Théorie des Nombres, Fasc. 1-2, Secrétariat mathématique, Paris, 1969, pp. Exp. 17, 7. MR 0244240
Jean-Paul Bézivin and Abdelbaki Boutabaa, Sur les équations fonctionelles $p$-adiques aux $q$-différences, Collect. Math. 43 (1992), no. 2, 125–140 (French, with English summary). MR 1223416
Jean-Paul Bézivin, Convergence des solutions formelles de certaines équations fonctionnelles, Aequationes Math. 44 (1992), no. 1, 84–99 (French, with English summary). MR 1165786, DOI 10.1007/BF01834207
Jean-Paul Bézivin, Sur les équations fonctionnelles aux $q$-différences, Aequationes Math. 43 (1992), no. 2-3, 159–176 (French, with English summary). MR 1158724, DOI 10.1007/BF01835698
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. MR 1288467, DOI 10.1007/BF01231526
Lucia Di Vizio, Arithmetic theory of $q$-difference equations: the $q$-analogue of Grothendieck-Katz’s conjecture on $p$-curvatures, Invent. Math. 150 (2002), no. 3, 517–578. MR 1946552, DOI 10.1007/s00222-002-0241-z
Lucia Di Vizio. Local analytic classification of $q$-difference equations with $|q|=1$. arXiv:0802.4223
L. Di Vizio, J.-P. Ramis, J. Sauloy, and C. Zhang, Équations aux $q$-différences, Gaz. Math. 96 (2003), 20–49 (French). MR 1988639
Monique Fleinert-Jensen, Théorèmes d’indices précisés et convergences des solutions pour une équation linéaire aux $q$-différences, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 4, 425–428 (French, with English and French summaries). MR 1351090
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. MR 2172488, DOI 10.2140/gtm.2004.7.291
K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, and Y. Yamada, Construction of hypergeometric solutions to the $q$-Painlevé equations, Int. Math. Res. Not. 24 (2005), 1441–1463. MR 2153786, DOI 10.1155/IMRN.2005.1439
Edmond Maillet. Sur les séries divergentes et les équations différentielles. Annales Scientifiques de l’École Normale Supérieure. Troisième Série, 20, 1903.
Bernard Malgrange, Sur le théorème de Maillet, Asymptotic Anal. 2 (1989), no. 1, 1–4 (French). MR 991413
Fabienne Naegele, Théorèmes d’indices pour les équations $q$-différences-différentielles, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 6, 579–582 (French, with English and French summaries). MR 1240803
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). MR 542737
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. MR 1838017, DOI 10.1016/S0898-1221(01)00180-8
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. MR 2179691
Yasutaka Sibuya and Steven Sperber, Convergence of power series solutions of $p$-adic nonlinear differential equation, Recent advances in differential equations (Trieste, 1978) Academic Press, New York-London, 1981, pp. 405–419. MR 643150
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. MR 606047
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. MR 604044, DOI 10.2307/1971135
Changgui Zhang, Sur un théorème du type de Maillet-Malgrange pour les équations $q$-différences-différentielles, Asymptot. Anal. 17 (1998), no. 4, 309–314 (French, with English and French summaries). MR 1656811