Asymptotic Differential Algebra and Model Theory of Transseries


Asymptotic differential algebra aims at understanding the asymptotics of solutions to differential equations from an algebraic point of view. This area includes the study of Hardy fields and is at the crossroads of algebra, analysis, and logic. The differential field of transseries plays a central role in the subject. Over the last thirty years, transseries emerged in various different ways: 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.

In this book we substantiate the intuition that the differential field of transseries is a universal domain for asymptotic differential algebra by establishing a complete elimination theory for systems of algebraic differential equations with asymptotic side conditions.

Authors: M. Aschenbrenner, L. van den Dries, J. van der Hoeven

Keywords: H-field, differential algebra, transseries, valuation, Hardy field, quantifier elimination, model completeness

Buy: Princeton University Press

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