\(p\)-adic sets of range uniqueness (Q2467955)

Assume that \(K\) is an algebraically closed field that is complete relative to an ultrametric absolute value \(| \;| \). Let also \(D\) be a subset of \(K\) and \(R(D)\) the set of rational functions defined on \(D\), and without poles in \(D\). By definition, the set \(H(D)\) of analytic elements on \(D\) is the completion of \(R(D)\) with respect to the topology \(U _ D\) of uniform convergence on \(D\). The set \(D\) is called infraconnected, if for each \(a \in D\), the image of the mapping \(I _ a\) of \(D\) into the set \(\mathbb{R} _ +\) of real nonnegative numbers, defined by the rule \(I _ a (x) = | x-a| \) has an image whose closure in \(\mathbb{R} _ +\) is an interval. A subset \(S \subseteq D\) is said to be a set of range uniqueness (SRU) for elements of \(H(D)\), in case for every \(f, g \in H(D)\) with \(f(s) = g(s)\), \(s \in S\), we have \(f = g\). The main results of the paper under review are obtained under the hypothesis that \(D\) is an infraconnected open and closed subset of \(K\) with no \(T\)-filter (in the sense of \textit{A. Escassut} [Ann. Inst. Fourier 25, 45--80 (1975; Zbl 0302.43012)]). The first one gives a necessary and sufficient condition for a monotonic distances sequence of diameter \(r > 0\) and centre \(a\) to be an SRU for \(H(D)\). The second one gives such a condition for a decreasing distances sequence of diameter \(r > 0\) without a centre in \(K\). This allows the author to find monotonic distances sequences which are SRUś substantially different from those known in the special case where \(K\) is the field of complex numbers. At the same time, it becomes clear that most of open closed subsets of \(K\) cannot be SRU. For example, this applies to open closed subsets with a hole, and to those which are not infraconnected.