Relative differential closure in Hardy fields

TL;DR

This paper proves a conjecture about Hardy fields by analyzing their relative differential closure, extending previous work on algebraic differential equations, and clarifies the limits of certain methods.

Contribution

It establishes the equivalence of intersections of maximal analytic and Hardy fields and generalizes a crucial proof component.

Findings

Confirmed the conjecture of Boshernitzan (1981).

Extended the understanding of algebraic differential equations over Hardy fields.

Provided a cautionary example illustrating the method's limitations.

Abstract

We study relative differential closure in the context of Hardy fields. Using our earlier work on algebraic differential equations over Hardy fields, this leads to a proof of a conjecture of Boshernitzan (1981): the intersection of all maximal analytic Hardy fields agrees with that of all maximal Hardy fields. We also generalize a key ingredient in the proof, and describe a cautionary example delineating the boundaries of its applicability.