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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 deals with eventual quantities, immediate extensions, and special cuts. It first considers the behavior of eventual quantities before discussing Newton weight, Newton degree, and Newton multiplicity as well as Newton weight of linear differential operators. It then establishes the following result: Every asymptotically maximal H-asymptotic field with rational asymptotic integration is spherically complete. The chapter proceeds by describing special (definable) cuts in H-asymptotic fields K with asymptotic integration and introducing some key elementary properties of K, namely λ‎-freeness and ω‎-freeness, which indicate that these cuts are not realized in K. It shows that has these properties. Finally, it looks at certain special existentially definable subsets of Liouville closed H-fields K, along with the behavior of the functions ω‎ and λ‎ on these sets.

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 considers the theory Tsuperscript nl of ω‎-free newtonian Liouville closed H-fields that eliminates quantifiers in a certain natural language. This theory has two completions: in the first, the models are the models of Tsuperscript nl with small derivation; in the second, the derivation is not small. One can move from models of the first completion to models of the second completion by compositional conjugation. The chapter begins with a discussion of extensions controlled by asymptotic couples and then shows the uniqueness-up-to-isomorphism of Newton-Liouville closures of ω‎-free H-fields. It then constructs a ω‎-free ΔΩ‎-field extension of K with a useful semiuniversal property. It also deduces Theorem 7 about quantifier elimination with various interesting consequences and concludes by specifying the language of ΔΩ‎-fields and demonstrating the elimination of quantifiers with applications.