Synopsis
After a construction of the complete ultrametric fields K, the book presents most of properties of analytic and meromorphic functions in K: algebras of analytic elements, power series in a disk, order, type and cotype of growth of entire functions, clean functions, question on a relation true for clean functions, and a counter-example on a non-clean function. Transcendence order and transcendence type are examined with specific properties of certain p-adic numbers. The Kakutani problem for the "corona problem" is recalled and multiplicative semi-norms are described. Problems on exponential polynomials, meromorphic functions are introduced and the Nevanlinna Theory is explained with its applications, particularly to problems of uniqueness. Injective analytic elements and meromorphic functions are examined and characterized through a relation. The Nevanlinna Theory out of a hole is described. Many results on zeros of a meromorphic function and its derivative are examined, particularly the solution of the Hayman conjecture in a P-adic field is given. Moreover, if a meromorphic functions in all the field, admitting primitives, admit a Picard value, then it must have enormously many poles. Branched values are examined, with links to growth order of the denominator. The Nevanlinna theory on small functions is explained with applications to uniqueness for a pair of meromorphic functions sharing a few small functions. A short presentation in characteristic p is given with applications on Yoshida equation.
About the Author
Alain Escassut obtained a Doctorate in 1970 and a Doctorat d'Etat in 1972 at Université Bordeaux 1 and worked there from 1969 to 1987 (he was Visiting-Assistant Professor at Princeton University in Spring 1981). Since 1987, he has been Professor at Université Blaise Pascal, (now Université Clermont Auvergne) and he is now emeritus Professor. He is a specialists of ultrametric analysis and particularly of analytic functions in an algebraically closed ultrametric field and their applications such as analytic elements defined by Marc Krasner, p-adic Nevanlinna Theory and applications to ultrametric Banach algebras. He characterized properties of analytic elements after defining T-filters and showed their role in ultrametric Banach algebras. He examined properties of uniqueness in joint works with Bertin Diarra, Abdelbaki Boutabaa, William Cherry, C C Yang, Ta Thi Hoai An. Several other problems are examined like the Hayman Conjecture in a P-adic field, zeros of exponential polynomials in zero residue characteristic and problems on the growth of p-adic entire functions. He took part in most of the conferences on p-adic functional analysis.
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