Eventual Quantities, Immediate Extensions, and Special Cuts

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.