...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...

...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...

...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...