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Analytic elements in \(p\)-adic analysis. (English) Zbl 0933.30030

Singapore: World Scientific. x, 390 p. (1995).
Hensel’s construction of the \(p\)-adic numbers in 1897 was motivated by number theoretical applications. Its essence was the compactness of the \(p\)-adic integers. In “Zahlentheorie” (1913) he defined \(p\)-adic exponential and logarithmic functions and proved their fundamental properties.
W. Schöbe in “Beiträge zur Funktionentheorie in nichtarchimedisch bewerteten Körpern” [Diss. Univ. Münster, (1930; JFM 56.0141.03)] started a systematic study of the function theory in such fields. L. Schnirelmann (L. Šnirel’man) [Bull. Acad. Sci. URSS, Ser. Math. 1938, No. 5/6, 487-498 (1938; JFM 64.1047.03)] initiated an approach to entire functions in algebraically closed non-Archimedean complete fields. His research was continued in F. Loonstra’s “Analytische Untersuchungen über bewertete Körper” [Diss. Univ. Amsterdam (1941)]. J. Dieudonné [Bull. Soc. Math., II. Ser. 68, 79-95 (1944; Zbl 0060.08204)] proved approximation theorems for continuous functions and studied a problem of primitive functions in the \(p\)-adic case. However, a systematic study of non-Archimedean analytic functions, analytical continuation etc. was initiated by M. Krasner in 1946 and described in details in 1966 (ss [48] in Escassut’s book). W. H. Schikhof presented in “Non-Archimedean harmonic analysis” [Disser. Univ. Nijmegen (1967; Zbl 0154.15401)] a systematic study of the \(p\)-adic Fourier transform. C. F. Woodcock extended his results in the seventieths.
The book under review presents in detail a state of the theory up to 1995. The results in non-Archimedean analysis can be divided into two parts:
(1) results motivated by concrete applications, mainly in the number theory and algebraic geometry; recently in physics based on \(p\)-adic models of the space;
(2) analogies of classical function theory and functional analysis in non-Archimedean fields and vector spaces.
The present book belongs to the second direction and discusses among others various aspects of the algebra \(H(D)\) constructed as the uniform completion of the algebra of rational functions defined on an infinite set \(D\subset K\) of an algebraically closed complete non-Archimedean field \(K\) under successive specializations of \(D\). Contemporary electronic technics give possibility of building new texts from the old ones. Many chapters of the book are almost identical with some of the author’s numerious papers: the chapters are reproduced from his original papers. E.g. Chapter 39 is almost the same as [31], Chapter 40 as [6], Chapter 56 as [26], etc. Consequently notations are changing from chapter to another one. I am not writing an editorial review, so I will only indicate some different remarks. The fundamental field is denoted in some chapters as \(L\), in another as \(K\). It is not clear what is assumed on \(K\): it was not stated in Chapters 5, 9-18, 20-27, 29-54.
The introduction to \(p\)-adic and non-Archimedean fields is too short for non-specialist readers, in particular, for students. Fundamental results on \(p\)-adic fields, such as extensions of absolute values, exponential and logarithmic functions, Hensel’s lemma, Krasner’s lemma etc. are scattered in different chapters. For example, infraconnected sets, monotonous and circular filters, valuations on \(K(x)\) (\(x\)-transcendental) are introduced in Chapters 2-4. Hensel’s lemma is proved in Chapter 5, extensions of valuations are discussed in Chapter 6, theorems on ultraproducts (\(Ax\)-Cochen) are proved in Chapter 7, and the \(p\)-adic numbers are suddenly introduced in Chapter 8!
It seems that it would be better for readers of the book, at first defined \(p\)-adic numbers, their properties, extensions of absolute valued fields, Hensel’s lemma, Krasner’s lemma and next more general results on function algebras and the whole theory of analytical functions. There are numerous errors, e.g. power-multiplicative norm is called semi-multiplicative (p. 3); \(K\)-linear would be read as \(L\)-linear (p. 3); \(\text{int}(x)\in\mathbb{Z}\) not \(\mathbb{N}\) (p. 8), etc. The term “Gauss norm” (p. 27) has nothing to do with Gauss. The introducing terminology would have some relations to historical achievements. In this case I do not see why Gauss. The English language of the book is far from being perfect. There is no algorithm for numbering formulae, e.g. in Chapter 39 one has formulas (1)–(4), (1)–(14) and again (1)–(3). The same happens in other places. The list of references is not complete. Only selected papers are quoted, mostly of French authors’. Many papers of Y. Amice, E. Motzkin, Ph. Robba, M.-C. Sarmant are omitted. The names such as G. Bachman, D. Barsky, S. Bosch, E. Beckenstein, U. Güntzer, L. Narici, R. Remmert, J.-P. Serre, C. F. Woodcock (and others) are omitted in the references. The above authors have papers and books closely related to \(p\)-adic analysis.
Many interesting theorems are proved in the book. E.g. a beautiful characterization of infraconnected sets is given (Theorem 19.5): a clopen set \(E\) is infraconnected, if and only if, the only function \(f\in H(E)\) with zero derivative is a constant function. (It is well-known that there exist many non-constant functions in \(p\)-adic analysis with zero derivative everywhere).
Summarizing, the book under review presents recent trends in \(p\)-adic analysis and will be useful for specialists, but less for students, contrary to the author’s statement in the introduction. Consequently, the book could be improved to a certain extent.

MSC:

30G06 Non-Archimedean function theory
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
12H25 \(p\)-adic differential equations
26E30 Non-Archimedean analysis
12J27 Krasner-Tate algebras
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
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