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This book examines ultrametric Banach algebras in general. It
begins with algebras of continuous functions, and looks for maximal
and prime ideals in connections with ultrafilters on the set of
definition. The multiplicative spectrum has shown to be
indispensable in ultrametric analysis and is described in the
general context and then, in various cases of Banach
algebras.Applications are made to various kind of functions:
uniformly continuous functions, Lipschitz functions, strictly
differentiable functions, defined in a metric space. Analytic
elements in an algebraically closed complete field (due to M
Krasner) are recalled with most of their properties linked to
T-filters and applications to their Banach algebras, and to the
ultrametric holomorphic functional calculus, with applications to
spectral properties. The multiplicative semi-norms of Krasner
algebras are characterized by circular filters with a metric and an
order that are examined.The definition of the theory of affinoid
algebras due to J Tate is recalled with all the main algebraic
properties (including Krasner-Tate algebras). The existence of
idempotents associated to connected components of the
multiplicative spectrum is described.
P-adic Analytic Functions describes the definition and properties
of p-adic analytic and meromorphic functions in a complete
algebraically closed ultrametric field.Various properties of p-adic
exponential-polynomials are examined, such as the Hermite-Lindemann
theorem in a p-adic field, with a new proof. The order and type of
growth for analytic functions are studied, in the whole field and
inside an open disk. P-adic meromorphic functions are studied, not
only on the whole field but also in an open disk and on the
complemental of an open disk, using Motzkin meromorphic products.
Finally, the p-adic Nevanlinna theory is widely explained, with
various applications. Small functions are introduced with results
of uniqueness for meromorphic functions. The question of whether
the ring of analytic functions-in the whole field or inside an open
disk-is a Bezout ring is also examined.
The book first explains the main properties of analytic functions
in order to use them in the study of various problems in p-adic
value distribution. Certain properties of p-adic transcendental
numbers are examined such as order and type of transcendence, with
problems on p-adic exponentials. Lazard's problem for analytic
functions inside a disk is explained. P-adic meromorphics are
studied. Sets of range uniqueness in a p-adic field are examined.
The ultrametric Corona problem is studied. Injective analytic
elements are characterized. The p-adic Nevanlinna theory is
described and many applications are given: p-adic Hayman
conjecture, Picard's values for derivatives, small functions,
branched values, growth of entire functions, problems of
uniqueness, URSCM and URSIM, functions of uniqueness, sharing value
problems, Nevanlinna theory in characteristic p>0, p-adic
Yosida's equation.
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