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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 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.

Abstract. Let A be a commutative Noetherian ring, and let R = A[X] be the polynomial ring in an infinite collection X of indeterminates over A. Let S X be the group of permutations of X. The group S X acts on R in a natural way, and this in turn gives R the structure of a left module over the group ring R[S X ]. We prove that all ideals of R invariant under the action of S X are finitely generated as R[S X ]-modules. The proof involves introducing a certain well-quasi-ordering on monomials and developing a theory of Gröbner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finite-dimensional polynomial rings and relate it to the finite generation result mentioned above. Finally, a motivating question from chemistry is presented, with the above framework providing a suitable context in which to study it.

We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, and P -minimal theories.Our approach to Theorem 1.1, via definable types, was partly inspired by the use of Puiseux series in [11,81].Let ACVF denote the theory of (non-trivially) valued algebraically closed fields, in the ring language expanded by a predicate for the valuation divisibility. This has completions ACVF (0,0) (for residue characteristic 0), ACVF (0,p) (field characteristic 0, residue characteristic p), and ACVF (p,p) (field characteristic p). Because ACVF (0,0) is interpretable in RCVF, our methods give (non-optimal) density bounds for ACVF (0,0) (Corollary 6.3). However, they give no information on density in the theories ACVF (0,p) and ACVF (p,p) . The problems arise essentially because a definable set in 1-space in ACVF is a finite union of 'Swiss cheeses' but we have no way of choosing a particular Swiss cheese. This means that the definable types technique in our main tool (Theorem 5.7) breaks down. On the other hand, our methods do yield:Theorem 1.2. Suppose M = Q p is the field of p-adic numbers, construed as a firstorder structure in Macintyre's language L p . Then the VC density of every L p -formula ϕ(x; y) is at most 2|y| − 1.

We introduce H-fields as ordered differential fields of a certain kind. Hardy fields extending R, as well as the field of logarithmicexponential series over R are H-fields. We study Liouville extensions in the category of H-fields, as a step towards a model theory of H-fields. The main result is that an H-field has at most two Liouville closures.