Value distribution in \(p\)-adic analysis (Q2802920)

A. Escassut is one of the leading experts in non-Archimedean analysis, especially in the part of the latter dealing with non-Archimedean analytic functions and Banach algebras. His book [Analytic elements in \(p\)-adic analysis. Singapore: World Scientific (1995; Zbl 0933.30030)] is the only detailed exposition of the theory of non-Archimedean analytic continuation created by Krasner and developed by several authors including Escassut. He is the author of another good book [Ultrametric Banach algebras. Singapore: World Scientific (2003; Zbl 1026.46067)] devoted to non-Archimedean operator algebras (including algebras of analytic functions), a subject related to non-Archimedean function theory. NEWLINENEWLINENEWLINENEWLINE The main aim of the proposed new book is the exposition of a non-Archimedean version of the Nevanlinna theory of value distribution. However its contents are much wider, it covers many subjects from non-Archimedean theory of analytic functions, such as vanishing of analytic elements along filters, factorization of analytic elements, and the corona problem. The author gives an analytic exposition of various subjects regarding algebraic and transcendent extensions of the field of \(p\)-adic numbers. This material is valuable even irrespective of the theory of meromorphic functions. In fact, the reader should not forget that the non-Archimedean theory possesses many features different from the classical one -- the reason is a much more complicated algebraic and topological structure of non-Archimedean fields, and the author pays a due attention to such notions as spherical completeness and its role in analytic problems. NEWLINENEWLINENEWLINENEWLINE As for the Nevanlinna theory itself, the author not only gives analogues of the Main Fundamental Theorems, but writes a lot about various applications (uniqueness sets, exceptional values, functional equations, etc.). The case of positive characteristic is considered separately. Comparing with the book by \textit{P.-C. Hu} and \textit{C.-C. Yang} [Meromorphic functions over non-Archimedean fields. Dordrecht: Kluwer Academic Publishers (2000; Zbl 0984.30027)], I have to note that the latter is more specialized and does not contain much of the general material from non-Archimedean analysis mentioned above. Of course, there is a number of results in Escassut's book which appeared in the last 15 years. NEWLINENEWLINENEWLINENEWLINE There are some misprints in the book. The reader should be warned that the reference numbers, in particular in the Introduction, not always agree with those in the bibliography. It would be desirable to create a complete index instead of the existing list of definitions. NEWLINENEWLINENEWLINENEWLINE However as a whole, the book will be a very useful complement to the existing (in my view, not rich enough) literature on non-Archimedean analysis.