Asymptotic Differential Algebra and Model Theory of Transseries
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A01=Joris van der Hoeven
A01=Lou van den Dries
A01=Matthias Aschenbrenner
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Algebraic closure
Algebraic differential equation
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Algebraic extension
Algebraic independence
Algebraic theory
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Annihilator (ring theory)
Asymptote
Asymptotic analysis
Asymptotic expansion
Author_Joris van der Hoeven
Author_Lou van den Dries
Author_Matthias Aschenbrenner
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Descriptive set theory
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Differential algebra
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Product details
- ISBN 9780691175430
- Weight: 1219g
- Dimensions: 152 x 235mm
- Publication Date: 06 Jun 2017
- Publisher: Princeton University Press
- Publication City/Country: US
- Product Form: Paperback
- Language: English
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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.
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).
