In 1923, Dulac published a proof of the claim that every real
analytic vector field on the plane has only finitely many limit cycles
(now known as Dulac's Problem). In the mid-1990s, Ilyashenko completed
Dulac's proof; his completion rests on the construction of a
quasianalytic class of functions. Unfortunately, this class has very few
known closure properties. For various reasons I will explain, we are
interested in constructing a larger quasianalytic class that is also a
Hardy field. This can be achieved using Ilyashenko's idea of superexact
asymptotic expansion. (Joint work with Zeinab Galal and Tobias Kaiser)
