# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc:1701.332024

Acta Mathematica, 214, (2), 307-355, 2014.5
We prove that the radii of convergence of the solutions of a$p$-adic differential equation $${\fancyscript{F}}$$ over an affinoid domain$X$of the Berkovich affine line are continuous functions on$X$that factorize through the retraction of $${X\to\Gamma}$$ of$X$onto a finite graph $${\Gamma\subseteq X}$$ . We also prove their super-harmonicity properties. This finiteness result means that the behavior of the radii as functions on$X$is controlled by a$finite$family of data.
Berkovich spaces; Radius of convergence; Newton polygon; spectral radius; controlling graph; finiteness
@inproceedings{andrea2014the,
title={The convergence Newton polygon of a$p$-adic differential equation I: Affinoid domains of the Berkovich affine line},
author={Andrea Pulita},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203405003739802},
booktitle={Acta Mathematica},
volume={214},
number={2},
pages={307-355},
year={2014},
}

Andrea Pulita. The convergence Newton polygon of a$p$-adic differential equation I: Affinoid domains of the Berkovich affine line. 2014. Vol. 214. In Acta Mathematica. pp.307-355. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203405003739802.