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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Complex analysis
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Meromorphic Functions over Non-Archimedean Fields (Paperback, Softcover reprint of hardcover 1st ed. 2000)
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Meromorphic Functions over Non-Archimedean Fields (Paperback, Softcover reprint of hardcover 1st ed. 2000)
Series: Mathematics and Its Applications, 522
Expected to ship within 10 - 15 working days
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Nevanlinna theory (or value distribution theory) in complex
analysis is so beautiful that one would naturally be interested in
determining how such a theory would look in the non Archimedean
analysis and Diophantine approximations. There are two "main
theorems" and defect relations that occupy a central place in N
evanlinna theory. They generate a lot of applications in studying
uniqueness of meromorphic functions, global solutions of
differential equations, dynamics, and so on. In this book, we will
introduce non-Archimedean analogues of Nevanlinna theory and its
applications. In value distribution theory, the main problem is
that given a holomorphic curve f : C -+ M into a projective variety
M of dimension n and a family 01 of hypersurfaces on M, under a
proper condition of non-degeneracy on f, find the defect relation.
If 01 n is a family of hyperplanes on M = r in general position and
if the smallest dimension of linear subspaces containing the image
f(C) is k, Cartan conjectured that the bound of defect relation is
2n - k + 1. Generally, if 01 is a family of admissible or normal
crossings hypersurfaces, there are respectively Shiffman's
conjecture and Griffiths-Lang's conjecture. Here we list the
process of this problem: A. Complex analysis: (i) Constant targets:
R. Nevanlinna[98] for n = k = 1; H. Cartan [20] for n = k > 1;
E. I. Nochka [99], [100],[101] for n > k ~ 1; Shiffman's
conjecture partially solved by Hu-Yang [71J; Griffiths-Lang's
conjecture (open).
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