Within the final 5 years there was very major growth within the improvement of transcendence concept. a brand new method of the mathematics houses of values of modular kinds and theta-functions was once chanced on. the answer of the Mahler-Manin challenge on values of modular functionality j(tau) and algebraic independence of numbers pi and e^(pi) are such a lot amazing result of this leap forward. The e-book offers those and different effects on algebraic independence of numbers and additional, an in depth exposition of equipment created in final the 25 years, within which commutative algebra and algebraic geometry exerted powerful catalytic impact at the improvement of the topic.

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