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The C.I.M.E. session in Diophantine Approximation, held in Cetraro (Italy) June 28 - July 6, 2000 focused on height theory, linear independence and transcendence in group varieties, Baker's method, approximations to algebraic numbers and applications to polynomial-exponential diophantine equations and to diophantine theory of linear recurrences. Very fine lectures by D. Masser, Y. Nesterenko, H.-P. Schlickewei, W.M. Schmidt and M. Walsschmidt have resulted giving a good overview of these topics, and describing central results, both classical and recent, emphasizing the new methods and ideas of the proofs rather than the details. They are addressed to a wide audience and do not require any prior specific knowledge.
In the last five years there has been very significant progress in
the development of transcendence theory. A new approach to the
arithmetic properties of values of modular forms and
theta-functions was found. The solution of the Mahler-Manin problem
on values of modular function j(tau) and algebraic independence of
numbers pi and e DEGREES(pi) are most impressive results of this
breakthrough. The book presents these and other results on
algebraic independence of numbers and further, a detailed
exposition of methods created in last the 25 years, during which
commutative algebra and algebraic geometry exerted strong catalytic
influence on the development of the s
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