Strictly analytic functions on \(p\)-adic analytic open sets (Q1301329)

Let \(K\) be an algebraically closed complete non-Archimedean valued field that is separable (for example the field \(\mathbb{C}_p\) of \(p\)-adic complex numbers). For an infinite set \(D\subset K\) the set \(H(D)\) of analytic elements on \(D\) is defined in the usual way. Recall that \(D\) is called analytic if, for every \(f\in H(D)\), for every disk \(\Delta\) with \(\Delta\cap D\neq\emptyset\), \(f= 0\) on \(\Delta\cap D\) implies \(f=0\). In the paper a new definition of an analytic function is proposed as follows. First, a new subclass of analytic sets, called analoids, is introduced, and it is shown that quasi-connected sets are analoids. Next, for an analoid \(D\), the class of the so-called \(D\)-admissible subsets of \(D\) is defined: they are certain clopen bounded analoids. The crucial property is the existence of an increasing sequence of \(D\)-admissible sets covering \(D\), which makes the following definition useful. A function \(f: D\to K\) is strictly analytic if for every \(D\)-admissible \(U\), the restriction of \(f\) to \(U\) is in \(H(U)\). Its importance lies in the properties (1) strictly analytic functions can be expanded in power (Laurent) series on every subdisk (subanulus) of \(D\), (2) the derivative of a strictly analytic function is strictly analytic. They show that this new theory overcomes some difficulties occurring in existing theories of Krasner-Robba, of Fresnel-van der Put, and of Karlowski and Ullrich.