Asymptotic Differential Algebra and Model Theory of Transseries (Annals of Mathematics Studies, 195)

Aschenbrenner, Matthias; Van Den Dries, Lou; Van Der Hoeven, Joris

ISBN 10: 0691175438 ISBN 13: 9780691175430

Editorial: Princeton University Press, 2017

Idioma: Ingl�s

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T�tulo

Asymptotic Differential Algebra and Model Theory of Transseries (Annals of Mathematics Studies, 195)

Editor

Princeton University Press

Fecha de publicaci�n

2017

Idioma

Ingl�s

ISBN 10

0691175438

ISBN 13

9780691175430

Encuadernaci�n

Encuadernaci�n de tapa blanda

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New

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Sinopsis

Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a central role in the subject. Besides powers of the variable, these series may contain exponential and logarithmic terms. Over the last thirty years, transseries emerged variously as super-exact asymptotic expansions of return maps of analytic vector fields, in connection with Tarski's problem on the field of reals with exponentiation, and in mathematical physics. Their formal nature also makes them suitable for machine computations in computer algebra systems. This self-contained book validates the intuition that the differential field of transseries is a universal domain for asymptotic differential algebra. It does so by establishing in the realm of transseries a complete elimination theory for systems of algebraic differential equations with asymptotic side conditions. Beginning with background chapters on valuations and differential algebra, the book goes on to develop the basic theory of valued differential fields, including a notion of differential-henselianity. Next, H-fields are singled out among ordered valued differential fields to provide an algebraic setting for the common properties of Hardy fields and the differential field of transseries. The study of their extensions culminates in an analogue of the algebraic closure of a field: the Newton-Liouville closure of an H-field. This paves the way to a quantifier elimination with interesting consequences.

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Matthias Aschenbrenner is professor of mathematics at the University of California, Los Angeles. Lou van den Dries is professor of mathematics at the University of Illinois, Urbana-Champaign. Joris van der Hoeven is director of research at the French National Center for Scientific Research (CNRS).

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