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arXiv:2408.05232v1 [math.LO] 2 Aug 2024
THE THEORY OF MAXIMAL HARDY FIELDS
MATTHIAS ASCHENBRENNER, LOU VAN DEN DRIES, AND JORIS VAN DER HOEVEN
Abstract. We show that all maximal Hardy fields are elementarily equivalent
as differential fields to the differential field Tof transseries, and give various
applications of this result and its proof.
Contents
Introduction 1
1. Preliminaries 13
2. Hardy Fields 18
3. Hardy Fields and Uniform Distribution 21
4. Universal Exponential Extensions of Hardy Fields 25
5. Inverting Linear Differential Operators over Hardy Fields 29
6. Solving Split-Normal Equations over Hardy Fields 36
7. Smoothness Considerations 41
8. Application to Filling Holes in Hardy Fields 44
9. Weights 48
10. Asymptotic Similarity 59
11. Differentially Algebraic Hardy Field Extensions 63
12. Transfer Theorems 69
13. Embeddings into Transseries and Maximal Hardy Fields 73
References 77
Introduction
Hardy [49] made sense of Du Bois-Reymond’s “orders of infinity” [17]–[20]. This led
to the notion of a Hardy field (Bourbaki [27]). A Hardy field is a field Hof germs
at +∞of differentiable real-valued functions on intervals (a, +∞) such that for any
differentiable function whose germ is in Hthe germ of its derivative is also in H.
(See Section 2 for more precision.) Every Hardy field is naturally a differential
field, and also an ordered field with the germ of fbeing >0 iff f(t)>0, eventually.
Hardy fields are the natural domain of asymptotic analysis, where all rules hold,
without qualifying conditions [72, p. 297]. Many basic facts about Hardy fields can
be found in Bourbaki [27], Boshernitzan [21]–[24], and Rosenlicht [72]–[75].
Hardy [47] focused on the Hardy field consisting of the germs of logarithmico-
exponential functions (LE-functions, for short): these functions are the real-valued
functions obtainable from real constants and the identity function xusing addition,
multiplication, division, taking logarithms, and exponentiating. Examples include
Date: August, 2024.
1
the germs of the function given for large positive xby xr(r∈R), ex2, and log log x.
Besides the germs of LE-functions, the germs of many other naturally occurring
non-oscillating differentially algebraic functions lie in Hardy fields. This includes
in particular several special functions like the error function erf, the exponential
integral Ei, the Airy functions Ai and Bi, etc. There are also Hardy fields which
contain (germs of) differentially transcendental functions, such as the Riemann ζ-
function and Euler’s Γ-function [72], and even functions ultimately growing faster
than each LE-function [23].
Germs of functions in Hardy fields are non-oscillating in a strong sense. In certain
applications, this kind of tameness alone is crucial: for example, ´
Ecalle’s proof [35]
of Dulac’s Conjecture (a weakened version of Hilbert’s 16th Problem) essentially
amounts to showing that the germ of the Poincar´e return map at a cross section
of a limit cycle lies in a Hardy field (at 0+ instead of +∞). A stronger form
of tameness is o-minimality: indeed, every o-minimal structure on the real field
naturally gives rise to a Hardy field (of germs of definable functions). This yields
a wealth of examples such as those obtained from quasi-analytic Denjoy-Carleman
classes [70], or containing certain transition maps of plane analytic vector fields [56],
and explains the role of Hardy fields in model theory and its applications to real
analytic geometry and dynamical systems [2, 14, 64].
Hardy fields have also found other applications: for effective counterparts to Hardy’s
theory of LE-functions, see [43, 46, 52, 67, 79]. Hardy fields have provided an
analytic setting for extensions of this work beyond LE-functions [50, 77, 78, 82, 83].
They have also been useful in ergodic theory (see, e.g., [13, 26, 42, 57]), and other
areas of mathematics [12, 28, 30, 37, 39, 45].
In the remainder of this introduction, His a Hardy field. Then H(R) (obtained
by adjoining the germs of the constant functions) is also a Hardy field, and for
any h∈H, the germ ehgenerates a Hardy field H(eh) over H, and so does any dif-
ferentiable germ with derivative h. Moreover, Hhas a unique Hardy field extension
that is algebraic over Hand real closed. (See Section 2 for these facts, especially
Proposition 2.1.)