The logarithmic growth of element of Robba ring which satisfies Frobenius equation over bounded Robba ring

TLDR: In this article, the authors studied the logarithmic growth of an element of the Robba ring which satisfies a Frobenius equation over bounded functions of rank 2 and showed that in special cases, the zeros of these functions have some cyclicity.

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.