Tohoku Mathematical Journal
2013
June
SECOND SERIES VOL. 65, NO. 2

Tohoku Math. J.
65 (2013), 179-198

Title THE LOGARITHMIC GROWTH OF ELEMENT OF ROBBA RING WHICH SATISFIES FROBENIUS EQUATION OVER BOUNDED ROBBA RING

Author Takahiro Nakagawa

(Received October 29, 2010, revised August 8, 2012)
Abstract. We study the logarithmic growth of an element of the Robba ring which satisfies a Frobenius equation over the bounded Robba ring. Chiarellotto and Tsuzuki computed the logarithmic growth of analytic functions on the open unit disc with coefficients in a $p$-adic local field which satisfy Frobenius equations over bounded functions of rank 2. We extend their result by replacing those functions by elements of the Robba ring which satisfy Frobenius equations over the bounded Robba ring. Moreover, we will see, in special cases, the zeros of these functions have some cyclicity and the logarithmic growth can be computed by the zeros of these function.

Mathematics Subject Classification. Primary 12H25.
Key words and phrases. Logarithmic growth, $p$-adic differential equation.

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