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March 2011 A note on non-Robba $p$-adic differential equations
Said Manjra
Proc. Japan Acad. Ser. A Math. Sci. 87(3): 40-43 (March 2011). DOI: 10.3792/pjaa.87.40


Let $\mathcal{M}$ be a differential module, whose coefficients are analytic elements on an open annulus $I$ ($\subset \mathbf{R}_{>0}$) in a valued field, complete and algebraically closed of inequal characteristic, and let $R(\mathcal{M}, r)$ be the radius of convergence of its solutions in the neighborhood of the generic point $t_{r}$ of absolute value $r$, with $r\in I$. Assume that $R(\mathcal{M}, r)<r$ on $I$ and, in the logarithmic coordinates, the function $r\longrightarrow R(\mathcal{M}, r)$ has only one slope on $I$. In this paper, we prove that for any $r\in I$, all the solutions of $\mathcal{M}$ in the neighborhood of $t_{r}$ are analytic and bounded in the disk $D(t_{r},R(\mathcal{M},r)^{-})$.


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Said Manjra. "A note on non-Robba $p$-adic differential equations." Proc. Japan Acad. Ser. A Math. Sci. 87 (3) 40 - 43, March 2011.


Published: March 2011
First available in Project Euclid: 3 March 2011

zbMATH: 1230.12007
MathSciNet: MR2802606
Digital Object Identifier: 10.3792/pjaa.87.40

Primary: 12H25

Keywords: $p$-adic differential equations , Frobenius antecedent theorem

Rights: Copyright © 2011 The Japan Academy

Vol.87 • No. 3 • March 2011
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