Asymptotic Fields and Asymptotic Couples

This chapter deals with asymptotic differential fields and their asymptotic couples. Asymptotic fields include Rosenlicht's differential-valued fields and share many of their basic properties. A key feature of an asymptotic field is its asymptotic couple. The chapter first defines asymptotic fields and their asymptotic couples before discussing H-asymptotic couples. It then considers asymptotic couples independent of their connection to asymptotic fields, along with the behavior of differential polynomials as functions on asymptotic fields. It also describes asymptotic fields with small derivation and the operations of coarsening and specialization, algebraic and immediate extensions of asymptotic fields, and differential polynomials of order one. Finally, it proves some useful extension results about asymptotic couples and establishes a property of closed H-asymptotic couples.