Asymptotic Differential Algebra and Model Theory of Transseries

Abstract

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 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.

Maximal immediate extensions of valued differential fields

“…In Section 7 we prove such uniqueness in the setting of valued differential fields, but only when the valuation is discrete. We also discuss there a conjecture from [2] about this, and recent progress on it.…”

“…We identify res(K) with res(K) via res(a) → res(a +Ȯ) for a ∈ O. The following is [2,Corollary 4.4.4]:…”

“…whose reduction P ∈ res(K){Y } has degree 1 has a zero in O; (cf. [2,Chapter 7]). If K is d-henselian, then its differential residue field res(K) is clearly linearly surjective.…”

Newtonian valued differential fields with arbitrary value group

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The notion of newtonianity is central to the study of the ordered differential field of logarithmic-exponential transseries done by Aschenbrenner, van den Dries, and van der Hoeven; see [1, Chapter 14]. We remove the assumption of divisible value group from two of their results concerning newtonianity, namely the newtonization construction and the equivalence of newtonianity with asymptotic differential-algebraic maximality.

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“…. 1 There is another generalization of henselianity to the valued differential field setting called "differentialhenselianity," introduced by Scanlon in [8] and developed in a more general setting in [1]. Although many asymptotic fields, including T, cannot be differential-henselian, there is a relationship between these two notions; see [1, §14.1].…”