Defining composition in abstract differential fields

Develop a precise and general definition of the composition f_1 \circ f_2 for elements f_1 and f_2 of an abstract differential field that preserves algebraic relations among derivatives of f_1 and is compatible with differentiation (i.e., (f_1^{(i)} \circ f_2)' = f_2' · (f_1^{(i+1)} \circ f_2) for all i).

Background

While composition of D‑algebraic functions is known to preserve D‑algebraicity when defined analytically (e.g., for meromorphic functions or power series), extending composition to elements viewed purely inside abstract differential fields is conceptually nontrivial.

The authors articulate desiderata for such a composition—preservation of algebraic relations and compatibility with the derivation rule—and then propose a working definition, highlighting that the general foundational question remains unclear.