The \(p\)-adic differential equation \(y'=\omega{} y\) in the closed unit disk (Q1182448)

Let \(\mathbb{K}\) be a complete ultrametric algebraically closed field, for instance \({\mathbb{K}}={\mathbb{C}}_ \rho\), and let \(H\) be the set of ``analytic elements in the closed unit disk \(| x|\leq 1\)'', namely the subset of \({\mathbb{K}}[[x]]\) made of those power series that converge in that disk. The authors give a characterization of the logarithmic differential \(\omega=f'/f\) of invertible functions \(f\in H\): they can be written as \(\omega(x)=\sum^ \infty_{n=1}(x-a_ n)^{-1}-(x-b_ n)^{-1}\) for sequences \(a_ n\) and \(b_ n\) in \(\mathbb{K}\) such that \(| b_ n|>1\), \(| a_ n|>1\), \(\lim| b_ n|=t\) and \(\lim(a_ n-b_ n)=0\). The particular case where \(\omega\) is constant is rather curious. Proofs are based on results about ``meromorphic products'' (i.e. infinite products of homographic functions) already obtained by both authors.