On natural radii of 𝑝-adic convergence

Dwork, B.; Robba, P.

doi:10.1090/S0002-9947-1979-0546915-9

On natural radii of $p$-adic convergence
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Trans. Amer. Math. Soc. 256 (1979), 199-213 Request permission

Abstract:

We study the radius of p-adic convergence of power series which represent algebraic functions. We apply the p-adic theory of ordinary linear differential equations to show that the radius of convergence is the natural one, provided the degree of the function is less than p. The study of similar questions for solutions of linear differential equations is indicated.

References

Scott Shorey Brown, Bounds on transfer principles for algebraically closed and complete discretely valued fields, Mem. Amer. Math. Soc. 15 (1978), no. 204, iv+92. MR 494980, DOI 10.1090/memo/0204
D. N. Clark, A note on the $p$-adic convergence of solutions of linear differential equations, Proc. Amer. Math. Soc. 17 (1966), 262–269. MR 186895, DOI 10.1090/S0002-9939-1966-0186895-8

Eléments algébriques

P. Dienes, The Taylor series: an introduction to the theory of functions of a complex variable, Dover Publications, Inc., New York, 1957. MR 0089895
B. Dwork and P. Robba, On ordinary linear $p$-adic differential equations, Trans. Amer. Math. Soc. 231 (1977), no. 1, 1–46. MR 447247, DOI 10.1090/S0002-9947-1977-0447247-8
Nicholas Katz, Travaux de Dwork, Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 409, Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973, pp. 167–200 (French, with English summary). MR 0498577

Untersuchungen zur arithmetischen Theorie der Körper

39

Philippe Robba, Fonctions analytiques sur les corps valués ultramétriques complets, Prolongement analytique et algèbres de Banach ultramétriques, Astérisque, No. 10, Société Mathématique de France, Paris, 1973, pp. 109–218, 219–220 (French, with English summary). MR 0357841
P. Robba, On the index of $p$-adic differential operators. I, Ann. of Math. (2) 101 (1975), 280–316. MR 364243, DOI 10.2307/1970992
O. F. G. Schilling, The Theory of Valuations, Mathematical Surveys, No. 4, American Mathematical Society, New York, N. Y., 1950. MR 0043776, DOI 10.1090/surv/004