# Ilyashenko algebras based on transserial asymptotic expansions

Our preprint extending my earlier construction of Ilyashenko algebras is now on the arXiv.

The purpose of this paper is to extend Ilyashenko’s construction of the class of germs at $+\infty$ of almost regular functions to obtain a Hardy field containing them.  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.  and a transseries as defined by van der Hoeven.  As these series generally have support of order type larger than $\omega$, the notion of asymptotic expansion itself needs to be generalized.  This can be done naturally in the context of a quasianalytic algebra, leading to our definition of \textit{quasianalytic asymptotic algebra}, or \textit{qaa algebra} for short.  Any qaa algebra constructed by generalizing Ilyashenko’s construction will be called an \textbf{Ilyashenko algebra}.
The Hardy field $\H = \Hanexp$ of all unary germs at $+\infty$ of unary functions definable in the o-minimal structure $\Ranexp$ is an example of an Ilyashenko field; see van den Dries and Miller  and van den Dries et al.  My earlier paper contains a first attempt at constructing an Ilyashenko field $\F$ containing Ilyashenko’s almost regular germs.  The implied non-oscillatory properties of its germs were used in Belotto et al.’s recent solution of the strong Sard conjecture.  However, this field $\F$ does not contain $\H$; the Ilyashenko field constructed here is a Hardy field that contains both $\F$ and $\H$, implying non-oscillatory behaviour with respect to all $\log$-$\exp$-analytic germs.
Our main motivation for generalizing Ilyashenko’s construction in this way is the conjecture that the class of almost regular germs generates an o-minimal structure over the field of real numbers.  This conjecture, in turn, might lead to locally uniform bounds on the number of limit cycles in subanalytic families of real analytic planar vector fields all of whose singularities are hyperbolic.  Establishing such uniform bounds for planar polynomial vector fields follows Roussarie’s approach to Hilbert’s 16th problem (part 2); see Ilyashenko’s survey for an overview on the latter.  Our conjecture implies a generic instance of Roussarie’s finite cyclicity conjecture; see my preprint explaining this connection. In Kaiser et al.  we gave a positive answer to our conjecture in the special case where all singularities are, in addition, non-resonant.  (For a different approach to the general hyperbolic case, see Mourtada’s preprint.)
The almost regular germs also play a role in the description of Riemann maps and solutions of Dirichlet’s problem on semianalytic domains; see Kaiser’s papers here and here  for details.  Finally, in the spirit of the concluding remark of van den Dries et al., this paper provides a rigorous construction of a Hardy subfield of Ecalle’s field of “fonctions analysables” that properly extends $\H$, and we do so without the use of “acc\’el\’ero-sommation”; for more details on this, see the concluding remarks in our paper.
We plan to eventually settle our o-minimality conjecture by adapting the procedure used here, which requires three main steps:
1. extend the class of almost regular germs into an Ilyashenko field;
2. construct corresponding algebras of germs of functions in several variables, such that the resulting system of algebras is stable under various operations (such as blowings-up, say);
3. obtain o-minimality using a normalization procedure.
While my earlier paper contains a first successful attempt at Step (1), Step (2) poses some challenges.  For instance, it is not immediately obvious what the nature of LE-series in several variables should be; they should at least be stable under all the operations required for Step (3).  They should also contain the series used in Mourtada’s preprint to characterize parametric transition maps in the hyperbolic case, which use so-called Ecalle-Roussarie compensators as monomials.
Our approach to this problem is to enlarge the set of monomials used in asymptotic expansions.  A first candidate for such a set of monomials is the set of all (germs of) functions definable in the o-minimal structure $\Ranexp$ (in any number of variables).  This set of germs is obviously closed under the required operations, because the latter are all definable, and it contains the Ecalle-Roussarie compensators.  However, it is too large to be meaningful for use as monomials in asymptotic expansions, as it is clearly not $\RR$-linearly independent (neither in the additive nor the multiplicative sense) and contains many germs that have “similar asymptotic behavior” such as, in the case of unary germs, belonging to the same archimedean class.  More suitable would be to find a minimal subclass $\la_n$ of all definable $n$-variable germs such that every definable $n$-variable germ is piecewise given by a convergent Laurent series (or, if necessary, a convergent generalized Laurent series in a finite tuple of germs in $\la_n$.
Thus, the purpose of this paper is to determine such a minimal set of monomials $\la = \la_1$ contained in the set $\H$ of all unary germs at $+\infty$ definable in $\Ranexp$, and to further adapt the construction in my earlier paper to corresponding generalized series in one variable.  Recalling that $\H$ is a Hardy field, we can summarize the results of this paper as follows:

### Main Theorem

There is a multiplicative subgroup $\la$ of $\H$ such that the following hold:
1. no two germs in $\la$ belong to the same archimedean class;
2. every germ in $\H$ is given by composing a convergent Laurent series with a tuple of germs in $\la$;
3. the construction in my earlier paper generalizes, after replacing the finite iterates of $\,\log$ with germs in $\la$, to obtain a corresponding Ilyashenko field $\K$.
The resulting Ilyashenko field $\K$ is a Hardy field extending $\H$ as well as the Ilyashenko field $\F$ constructed in my earlier paper.
We obtain this set $\la$ of monomials by giving an explicit description of the Hardy field $\H$ as the set of all convergent LE-series, with $\la$ being the corresponding set of \textit{convergent LE-monomials}.  The proof that the construction in my earlier paper generalizes to this set $\la$ relies heavily on our recent paper on $\log$-$\exp$-analytic germs; indeed, our construction here was our main motivation for this paper.

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