Ilyashenko algebras based on transserial asymptotic expansions

TL;DR

This paper constructs a Hardy field encompassing various classes of functions, characterized by unique asymptotic expansions using generalized LE-series, expanding the understanding of asymptotic analysis in transserial contexts.

Contribution

It introduces a Hardy field containing Ilyashenko's germs and log-exp-analytic germs, with uniquely characterized asymptotic expansions via generalized LE-series.

Findings

Constructed a Hardy field with Ilyashenko's germs and log-exp-analytic germs.

Established unique characterization of germs through generalized LE-series.

Extended asymptotic expansion concepts to support larger order types.

Abstract

We construct a Hardy field that contains Ilyashenko's class of germs at infinity of almost regular functions as well as all log-exp-analytic germs. In addition, each germ in this Hardy field is uniquely characterized by an asymptotic expansion that is an LE-series as defined by van den Dries et al. As these series generally have support of order type larger than that of the set of natural numbers, the notion of asymptotic expansion itself needs to be generalized.