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This chapter deals with differential polynomials. It first presents some basic facts about differential fields that are of characteristic zero with one distinguished derivation, along with their extensions. It then considers various decompositions of differential polynomials in their natural setting, along with valued differential fields and the property of continuity of the derivation with respect to the valuation topology. It also discusses the gaussian extension of the valuation to the ring of differential polynomials and concludes with some basic results on simple differential rings and differentially closed fields. In contrast to the corresponding notions for fields, the chapter shows that differential fields always have proper d-algebraic extensions, and the differential closure of a differential field K is not always minimal over K.

This chapter considers H-fields, pre-differential-valued fields with a field ordering that interacts with the valuation and derivation. Axiomatizing this interaction yields the notion of a pre-H-field; H-fields are d-valued pre-H-fields. The chapter begins by upgrading some basic facts on asymptotic fields to pre-d-valued fields; for example, algebraic extensions of pre-d-valued fields are pre-d-valued, not just asymptotic. It then adjoins integrals to pre-d-valued fields of H-type. It shows that every pre-d-valued field of H-type has a canonical differential-valued extension. It also adjoins exponential integrals to pre-d-valued fields of H-type. Finally, it describes Liouville closed H-fields, and especially the uniqueness properties of Liouville closure.

This book develops the algebra and model theory of the differential field of transseries, which are formal series in an indeterminate x > ℝ. is a field containing ℝ as a subfield and acquires the structure of a differential field. This introduction provides an overview of the importance of in different areas of mathematics such as analysis, computer algebra, and logic, as well as some of the most distinctive features of . In particular, it discusses the ordered and valued differential field , grid-based transseries, H-fields, closure properties, and valuations and asymptotic relations. It also explains the book's strategy and main results and considers a few more open-ended avenues of inquiry.

This chapter considers differential-henselian fields with many constants. Here d-henselian includes having small derivation, so d-henselian valued differential fields with many constants are monotone. The goal here is to derive Scanlon's extension in [382, 383] of the Ax-Kochen-Eršov theorems to d-henselian valued differential fields with many constants. Among the results to be established is Theorem 8.0.1: Suppose K and L are d-henselian valued differential fields with many constants. Then K = L as valued differential fields if and only if res K = res L as differential fields and Γ‎ subscript K = Γ‎ subscript L as ordered abelian groups. The chapter also discusses an angular component map on K, equivalence over substructures, relative quantifier elimination, and a model companion.

This chapter deals with valued differential fields, starting the discussion with an overview of the asymptotic behavior of the function vsubscript P: Γ‎ → Γ‎ for homogeneous P ∈ K K{Y}superscript Not Equal To. The chapter then shows that the derivation of any valued differential field extension of K that is algebraic over K is also small. It also explains how differential field extensions of the residue field k give rise to valued differential field extensions of K with small derivation and the same value group. Finally, it discusses asymptotic couples, dominant part, the Equalizer Theorem, pseudocauchy sequences, and the construction of canonical immediate extensions.

This chapter focuses on the Newton polynomial based on assumption that K is a differential-valued field of H-type with asymptotic integration and small derivation. Here K is also assumed to be equipped with a monomial group and (Γ‎, ψ‎) is the asymptotic couple of K. Throughout, P is an element of K{Y}superscript Not Equal To. The chapter first revisits the dominant part of P before discussing the elementary properties of the Newton polynomial. It then presents results about the shape of the Newton polynomial and considers realizations of three cuts in the value group Γ‎ of K. It also describes eventual equalizers, along with further consequences of ω‎-freeness and λ‎-freeness, the asymptotic equation over K, and some special H-fields.

This chapter discusses differential-henselian fields. Here K is a valued differential field with small derivation. An extension of K means a valued differential field extension of K whose derivation is small. After some preliminaries about d-henselianity, the chapter proves Theorem 7.0.1 stating that if k is linearly surjective and K is d-algebraically maximal, then K is d-henselian. For monotone K with linearly surjective k it proves the uniqueness-up-to-isomorphism-over-K of maximal immediate extensions. It also considers the case of few constants and shows that in the presence of monotonicity (perhaps unnecessary) a converse to Theorem 7.0.1 can be obtained. Finally, it describes differential-henselianity in several variables.

This chapter deals with valued abelian groups. It first introduces some terminology concerning ordered sets before discussing valued abelian groups and ordered abelian groups in more detail. Ordered abelian groups occur as value groups of valued fields, whereas valued abelian groups arise because the logarithmic derivative map on a valued differential field like induces a valuation on the value group that turns out to be very useful. Furthermore, the notion of a pseudocauchy sequence makes perfect sense in the general setting of valued abelian groups, and the basic facts about these sequences yield a natural proof of a generalized Hahn Embedding Theorem. The chapter also considers valued vector spaces, including spherically complete valued vector spaces, and proves a version of the Hahn Embedding Theorem for valued vector spaces. Special attention is given to particularly well-behaved valued vector spaces known as Hahn spaces.

This chapter considers the newtonianity of directed unions and proves an analogue of Hensel's Lemma for ω‎-free differential-valued fields of H-type: Theorem 15.0.1. Here K is an H-asymptotic field with asymptotic couple (Γ‎, ψ‎), and γ‎ ranges over Γ‎. The chapter first describes finitely many exceptional values, integration and the extension K(x), and approximating zeros of differential polynomials before proving Theorem 15.0.1, which states: If K is d-valued with ∂K = K, and K is a directed union of spherically complete grounded d-valued subfields, then K is newtonian. In concrete cases the hypothesis K = ∂K in the theorem can often be verified by means of Corollary 15.2.4.