Analyticity in spaces of convergent power series and applications (Q2801892)

The aim of this article is to study the analytic structure of the space \(\mathbb{C}\{z\}\) of germs of an analytic function at the origin of \(\mathbb{C}^m\) endowed with a particular locally convex topology induced by a convenient family of norms. The author proposes a framework in which the analytic properties of this space can easily be manipulated. Several properties of analytic sets of \(\mathbb{C}\{z\}\) as defined by the vanishing loci of analytic maps are investigated. In particular, it is shown that the countable union of proper analytic sets of \(\mathbb{C}\{z\}\) has empty interior. This property underlies the notion of a generic property of \(\mathbb{C}\{z\}\) for which some related theorems about complex dynamics are proved. Some applications of the main results to complex analysis and ordinary differential equations are included. The article continues and complements results by \textit{Y. Genzmer} and the author [J. Differ. Equations 248, No. 5, 1256--1267 (2010; Zbl 1192.34047)].