Matthias Aschenbrenner

Lou Van Den Dries

Joris Van Der Hoeven

Joris Van

On Numbers, Germs, and Transseries

Keywords: Mathematical Subject Classification. Primary 03C64, Secondary 12H05, 12J15 Date

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Tarski [START_REF] Tarski | A Decision Method for Elementary Algebra and Geometry[END_REF], an elementary extension of its ordered subfield of real numbers. (In fact, every set-sized real closed field embeds into the field of surreal numbers.) M. Kruskal anticipated the use of surreal numbers in asymptotics, and based on his ideas Gonshor [START_REF] Gonshor | An Introduction to the Theory of Surreal Numbers[END_REF] extended the exponential function on the reals to one on the surreals, with the same first-order logical properties [START_REF] Van Den Dries | Fields of surreal numbers and exponentiation[END_REF]. Rudiments of analysis for functions on the surreal numbers have also been developed [START_REF] Alling | Foundations of Analysis over Surreal Number Fields[END_REF][START_REF] Costin | Integration on the surreals: a conjecture of Conway[END_REF][START_REF] Rubinstein-Salzedo | Analysis on surreal numbers[END_REF].

Transseries generalize LE-functions in a similar way that surreals generalize reals and ordinals. Transseries have a precursor in the generalized power series of Levi-Civita [START_REF] Levi-Civita | Sugli infiniti ed infinitesimi attuali quali elementi analitici[END_REF][START_REF] Levi-Civita | Sui numeri transfiniti[END_REF] and Hahn [START_REF] Hahn | Über die nichtarchimedischen Größensysteme[END_REF], but were only systematically considered in the 1980s, independently by Écalle [START_REF] Écalle | Introduction aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac[END_REF] and Dahn-Göring [START_REF] Dahn | Notes on exponential-logarithmic terms[END_REF]. Écalle introduced transseries as formal counterparts to his "analyzable functions", which were central to his work on Dulac's Problem (related to Hilbert's 16th Problem on polynomial vector fields). Dahn and Göring were motivated by Tarski's Problem on the model theory of the real field with exponentiation. Transseries have since been used in various parts of mathematics and physics; their formal nature also makes them suitable for calculations in computer algebra systems. Key examples of transseries are the logarithmic-exponential series (LE-series for short) [START_REF] Van Den Dries | Logarithmic-exponential power series[END_REF][START_REF] Van Den Dries | Logarithmic-exponential series[END_REF]; more general notions of transseries have been introduced in [49,[START_REF] Schmeling | Corps de transséries[END_REF]. A transseries can represent a function of a real variable using exponential and logarithmic terms, going beyond the more prevalent asymptotic expansions in terms of powers of the independent variable. Transseries can be manipulated algebraically-added, subtracted, multiplied, divided-and like power series, can be differentiated termwise: they comprise a differential field. However, they carry much more structure: for example, by virtue of its construction, the field of LE-series comes with an exponential function; there is a natural notion of composition for transseries; and differential-compositional equations in transseries are sometimes amenable to functional-analytic techniques [START_REF] Van Der Hoeven | Operators on generalized power series[END_REF].

The logical properties of the exponential field of LE-series have been well-understood since the 1990s: by [START_REF] Wilkie | Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function[END_REF] and [START_REF] Van Den Dries | Logarithmic-exponential power series[END_REF] it is model-complete and o-minimal. In our book [START_REF] Aschenbrenner | Asymptotic Differential Algebra and Model Theory of Transseries[END_REF] we focused instead on the differential field of LE-series, denoted below by T, and obtained some decisive results about its model theory. Following A. Robinson's general ideas we placed T into a suitable category of H-fields and, by developing the extension theory of H-fields, showed that T is existentially closed as an H-field: each system of algebraic differential equations and inequalities over T which has a solution in an H-field extension of T already has one in T itself. In [START_REF] Aschenbrenner | Asymptotic Differential Algebra and Model Theory of Transseries[END_REF] we also prove the related fact that T is model-complete; indeed, we obtain a quantifier elimination (in a natural language) for T. As a consequence, the elementary theory of T is decidable, and model-theoretically "tame" in various ways: for example, it has Shelah's non-independence property (NIP).

Results from [START_REF] Aschenbrenner | Asymptotic Differential Algebra and Model Theory of Transseries[END_REF] about existential closedness, model completeness, and quantifier elimination substantiate the intuition, expressed already in [START_REF] Écalle | Introduction aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac[END_REF], that T plays the role of a universal domain for the part of asymptotic differential algebra that steers clear of oscillations. How far does this intuition lead us? Hardy's field of LE-functions embeds into T, as an ordered differential field, but this fails for other Hardy fields. The natural question here is: Are all maximal Hardy fields elementarily equivalent to T? It would mean that any maximal Hardy field instantiates Hardy's vision of a maximally inclusive and well-behaved algebra of oscillation-free real functions. Related is the issue of embedding Hardy fields into more general differential fields of transseries. Positive answers to these questions would tighten the link between germs of functions (living in Hardy fields) and their transseries expansions. We may also ask how surreal numbers fit into the picture: Is there a natural isomorphism between the field of surreal numbers and some field of generalized transseries? This would make it possible to differentiate and compose surreal numbers as if they were functions, and confirm Kruskal's premonition of a connection between surreals and the asymptotics of functions.

We believe that answers to these questions are within grasp due to advances in our understanding during the last decade as represented in our book [START_REF] Aschenbrenner | Asymptotic Differential Algebra and Model Theory of Transseries[END_REF]. We discuss these questions with more details in Sections 3, 4, 5. In Section 1 we set the stage by describing Hardy fields and transseries as two competing approaches to the asymptotic behavior of non-oscillatory real-valued functions. (Section 5 includes a brief synopsis of the remarkable surreal number system.) In Section 2 we define H-fields and state the main results of [START_REF] Aschenbrenner | Asymptotic Differential Algebra and Model Theory of Transseries[END_REF].

We let m, n range over N = {0, 1, 2, . . . }. Given an (additive) abelian group A we let A ̸ = := A \ {0}. In some places below we assume familiarity with very basic model theory, for example, on the level of [START_REF] Aschenbrenner | Asymptotic Differential Algebra and Model Theory of Transseries[END_REF]Appendix B]. "Definable" will mean "definable with parameters".

Orders of Infinity and Transseries

Germs of continuous functions

Consider continuous real-valued functions whose domain is a subset of R containing an interval (a, +∞), a ∈ R. Two such functions have the same germ (at +∞) if they agree on an interval (a, +∞), a ∈ R, contained in both their domains; this defines an equivalence relation on the set of such functions, whose equivalence classes are called germs. Addition and multiplication of germs is defined pointwise, giving rise to a commutative ring C. For a germ g of such a function we also let g denote that function if the resulting ambiguity is harmless. With this convention, given a property P of real numbers and g ∈ C we say that P g(t) holds eventually if P g(t) holds for all sufficiently large real t in the domain of g. We identify each real number r with the germ of the constant function R → R with value r. This makes the field R into a subring of C.

Following Hardy we define for f, g ∈ C,

f ≼ g : ⇐⇒ for some c ∈ R >0 we have |f (t)| ⩽ c|g(t)| eventually, f ≺ g : ⇐⇒ for every c ∈ R >0 we have |f (t)| < c|g(t)| eventually.
The reflexive and transitive relation ≼ yields an equivalence relation ≍ on C by setting f ≍ g :⇐⇒ f ≼ g and g ≼ f , and ≼ induces a partial ordering on the set of equivalence classes of ≍; these equivalence classes are essentially du Bois-Reymond's "orders of infinity". Thus with x the germ of the identity function on R:

0 ≺ 1 ≺ log log x ≺ log x ≺ √ x ≺ x ≍ -2x + x sin x ≺ x 2 ≺ e x .
One way to create interesting subrings of C is via expansions of the field of real numbers: any such expansion R gives rise to the subring H( R) of C consisting of the germs of the continuous functions R → R that are definable in R.

Hausdorff fields

A Hausdorff field is by definition a subfield of C. Simple examples are Q, R, R(x), R( √ x), R(x, e x , log x). (1.1)
That R(x, e x , log x) is a Hausdorff field, for instance, follows from two easy facts: first, an element f of C is a unit iff f (t) ̸ = 0 eventually (and then either f (t) > 0 eventually or f (t) < 0 eventually), and if f ̸ = 0 is an element of the subring R[x, e x , log x] of C, then f ≍ x k e lx (log x) m for some k, l, m ∈ N. Alternatively, one can use the fact that an expansion R of the field of reals is o-minimal iff H( R) is a Hausdorff field, and note that the examples above are subfields of H(R exp ) where R exp is the exponential field of real numbers, which is well-known to be o-minimal by Wilkie [START_REF] Wilkie | Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function[END_REF].

Let H be a Hausdorff field. Then H becomes an ordered field with (total) ordering given by: f > 0 iff f (t) > 0 eventually. Moreover, the set of orders of infinity in H is totally ordered by ≼: for f, g ∈ H we have f ≼ g or g ≼ f . In his landmark paper [START_REF] Hausdorff | Die Graduierung nach dem Endverlauf[END_REF], Hausdorff essentially proved that H has a unique algebraic Hausdorff field extension that is real closed. (Writing before Artin and Schreier [START_REF] Artin | Algebraische Konstruktion reeller Körper[END_REF], of course he doesn't use this terminology.) He was particularly interested in "maximal" objects and their order type. By Hausdorff's Maximality Principle (a form of Zorn's Lemma) every Hausdorff field is contained in one that is maximal with respect to inclusion. By the above, maximal Hausdorff fields are real closed. Hausdorff also observed that maximal Hausdorff fields have uncountable cofinality; indeed, he proved the stronger result that the underlying ordered set of a maximal Hausdorff field H is η 1 : if A, B are countable subsets of H and A < B, then A < h < B for some h ∈ H. A real closed ordered field is ℵ 1 -saturated iff its underlying ordered set is η 1 . Standard facts from model theory (or [START_REF] Erdős | An isomorphism theorem for real-closed fields[END_REF]) now yield an observation that could have been made by Hausdorff himself in the wake of Artin and Schreier [START_REF] Artin | Algebraische Konstruktion reeller Körper[END_REF]: Corollary 1.1. Assuming CH (the Continuum Hypothesis), all maximal Hausdorff fields are isomorphic. This observation was in fact made by Ehrlich [START_REF] Ehrlich | The absolute arithmetic continuum and the unification of all numbers great and small[END_REF] in the more specific form that under CH any maximal Hausdorff field is isomorphic to the field of surreal numbers of countable length; see Section 5 below for basic facts on surreals. We don't know whether here the assumption of CH can be omitted. (By [START_REF] Esterle | Solution d'un problème d'Erdös, Gillman et Henriksen et application à l'étude des homomorphismes de C(K)[END_REF], the negation of CH implies the existence of non-isomorphic real closed η 1 -fields of size 2 ℵ0 .) It may also be worth mentioning that the intersection of all maximal Hausdorff fields is quite small: it is just the field of real algebraic numbers.

Hardy fields

A Hardy field is a Hausdorff field whose germs can be differentiated. This leads to a much richer theory. To define Hardy fields formally we introduce the subring

C n := {f ∈ C : f is eventually n times continuously differentiable} of C, with C 0 = C. Then each f ∈ C n+1 has derivative f ′ ∈ C n . A Hardy field is a subfield of C 1 that is closed under f → f ′ ;
Hardy fields are thus not only ordered fields but also differential fields. The Hausdorff fields listed in (1.1) are all Hardy fields; moreover, for each o-minimal expansion R of the field of reals, H( R) is a Hardy field. As with Hausdorff fields, each Hardy field is contained in a maximal one. For an element f of a Hardy field we have either f ′ > 0, or f ′ = 0, or f ′ < 0, so f is either eventually strictly increasing, or eventually constant, or eventually strictly decreasing. (This may fail for f in a Hausdorff field.) Each element of a Hardy field is contained in the intersection n C n , but not necessarily in its subring C ∞ consisting of those germs which are eventually infinitely differentiable. In a Hardy field H, the ordering and derivation interact in a pleasant way: if f ∈ H and f > R, then f ′ > 0. Asymptotic relations in H can be differentiated and integrated: for 0

̸ = f, g ̸ ≍ 1 in H, f ≼ g iff f ′ ≼ g ′ .

Extending Hardy fields

Early work on Hardy fields focussed on solving algebraic equations and simple first order differential equations: Borel [START_REF] Borel | Mémoire sur les séries divergentes[END_REF], Hardy [START_REF] Hardy | Properties of logarithmico-exponential functions[END_REF][START_REF] Hardy | Some results concerning the behaviour at infinity of a real and continuous solution of an algebraic differential equation of the first order[END_REF], Bourbaki [START_REF] Bourbaki | Élements de Mathématique[END_REF], Marić [START_REF] Marić | Asymptotic behavior of solutions of a nonlinear differential equation of the first order[END_REF], Sjödin [START_REF] Sjödin | Hardy-fields[END_REF], Robinson [START_REF] Robinson | On the real closure of a Hardy field[END_REF], Rosenlicht [START_REF] Rosenlicht | Hardy fields[END_REF]. As a consequence, every Hardy field H has a smallest real closed Hardy field extension Li(H) ⊇ R that is also closed under integration and exponentiation; call Li(H) the Hardy-Liouville closure of H. (Hardy's field of LE-functions mentioned earlier is contained in Li(R).) Here is a rather general result of this kind, due to Singer [START_REF] Singer | Asymptotic behavior of solutions of differential equations and Hardy fields: preliminary report[END_REF]: Theorem 1.2. If y ∈ C 1 satisfies a differential equation y ′ P (y) = Q(y) where P (Y ) and Q(Y ) are polynomials over a Hardy field H and P (y) is a unit in C, then y generates a Hardy field H⟨y⟩ = H(y, y ′ ) over H.

Singer's theorem clearly does not extend to second order differential equations: the nonzero solutions of y ′′ + y = 0 in C 2 do not belong to any Hardy field. The solutions in C 2 of the differential equation

y ′′ + y = e x 2 (1.2)
form a two-dimensional affine space y 0 + R sin x + R cos x over R, with y 0 any particular solution. Boshernitzan [START_REF] Boshernitzan | Second order differential equations over Hardy fields[END_REF] proved that any of these continuum many solutions generates a Hardy field. Since no Hardy field can contain more than one solution, there are at least continuum many different maximal Hardy fields. By the above, each of them contains R, is real closed, and closed under integration and exponentiation. What more can we say about maximal Hardy fields?

To give an answer to this question, consider the following conjectures about Hardy fields H:

A. For any differential polynomial P (Y ) ∈ H{Y } = H[Y, Y ′ , Y ′′ , .
. . ] and f < g in H with P (f ) < 0 < P (g) there exists y in a Hardy field extension of H such that f < y < g and P (y) = 0.

B. For any countable subsets A < B in H there exists y in a Hardy field extension of H such that A < y < B.

Conjecture A for P ∈ H[Y, Y ′ ] holds by [START_REF] Van Den Dries | An intermediate value property for first-order differential polynomials[END_REF]. Conjecture A implies that all maximal Hardy fields are elementarily equivalent as we shall see in in Section 2. Conjecture B was first raised as a question by Ehrlich [START_REF] Ehrlich | The absolute arithmetic continuum and the unification of all numbers great and small[END_REF]. The conjectures together imply that, under CH, all maximal Hardy fields are isomorphic (the analogue of Corollary 1.1). We sketch a program to prove A and B in Section 3.

Transseries

Hardy made the point that the LE-functions seem to cover all orders of infinity that occur naturally in mathematics [42, p. 35]. But he also suspected that the order of infinity of the compositional inverse of (log x)(log log x) differs from that of any LEfunction [START_REF] Hardy | Properties of logarithmico-exponential functions[END_REF]; this suspicion is correct. For a more revealing view of orders of infinity and a more comprehensive theory we need transseries. For example, transseries lead to an easy argument to confirm Hardy's suspicion [START_REF] Van Den Dries | Logarithmic-exponential power series[END_REF]49]. Here we focus on the field T of LE-series and in accordance with [START_REF] Aschenbrenner | Asymptotic Differential Algebra and Model Theory of Transseries[END_REF], simply call its elements transseries, bearing in mind that many variants of formal series, such as those appearing in [START_REF] Schmeling | Corps de transséries[END_REF] (see Section 4 below), can also rightfully be called "transseries".

Transseries are formal series f = m f m m where the f m are real coefficients and the m are "transmonomials" such as

x r (r ∈ R), x -log x , e x 2 e x
, e e x .

One can get a sense by considering an example like 7 e e x + e x/2 + e x/4

+••• -3 e x 2 +5x √ 2 -(log x) π + 42 + x -1 + x -2 + • • • + e -x .
Here think of x as positive infinite: x > R. The transmonomials in this series are arranged from left to right in decreasing order. The reversed order type of the set of transmonomials that occur in a given transseries can be any countable ordinal. (In the example above it is ω + 1 because of the term e -x at the end.) Formally, T is an ordered subfield of a Hahn field R[[G LE ]] where G LE is the ordered group of transmonomials (or LE-monomials). More generally, let M be any (totally) ordered commutative group, multiplicatively written, the m ∈ M being thought of as monomials, with the ordering denoted by ≼.

The Hahn field R[[M]] consists of the formal series f = m f m m with real coefficients f m whose support supp f := {m ∈ M : f m ̸ = 0} is well-based , that is, well-ordered in the reversed ordering ≽ of M.
Addition and multiplication of these Hahn series works just as for ordinary power series, and the ordering of R[[M]] is determined by declaring a nonzero Hahn series to be positive if its leading coefficient is positive (so the series above, with leading coefficient 7, is positive). Both R[[G LE ]] and its ordered subfield T are real closed. Informally, each transseries is obtained, starting with the powers x r (r ∈ R), by applying the following operations finitely many times:

(1) multiplication with real numbers;

(2) infinite summation in R[[G LE ]];

(3) exponentiation and taking logarithms of positive transseries.

To elaborate on (2), a family

(f i ) i∈I in R[[M]
] is said to be summable if for each m there are only finitely many i ∈ I with m ∈ supp f i , and i∈I supp f i is wellbased; in this case we define the

sum f = i∈I f i ∈ R[[M]
] of this family by f m = i∈I (f i ) m for each m. One can develop a "strong" linear algebra for this notion of "strong" (infinite) summation [START_REF] Van Der Hoeven | Transseries and Real Differential Algebra[END_REF][START_REF] Schmeling | Corps de transséries[END_REF]. As to (3), it may be instructive to see how to exponentiate a transseries f : decompose f as f = g + c + ε where g := m≻1 f m m is the infinite part of f , c := f 1 is its constant term, and ε its infinitesimal part (in our example c = 42 and ε

= x -1 + x -2 + • • • + e -x ); then e f = e g • e c • n ε n n! where e g ∈ M is a transmonomial, and e c ∈ R, n ε n n! ∈ R[[G LE ]]
have their usual meaning. The story with logarithms is a bit different: taking logarithms may also create transmonomials, such as log x, log log x, etc.

The formal definition of T is inductive and somewhat lengthy; see [START_REF] Van Den Dries | Logarithmic-exponential series[END_REF][START_REF] Edgar | Transseries for beginners[END_REF][START_REF] Van Der Hoeven | Transseries and Real Differential Algebra[END_REF]] or [4, Appendix A] for detailed expositions. We only note here that by virtue of the construction of T, series like 1

x

+ 1 e x + 1 e e x + • • • or 1 x + 1 x log x + 1 x log x log log x + • • • (involving "nested" exponentials or logarithms of unbounded depth), though they are legitimate elements of R[[G LE ]],
do not appear in T; moreover, the sequence x, e x , e e x , . . . is cofinal in T, and the sequence x, log x, log log x, . . . is coinitial in the set {f ∈ T : f > R}. The map f → e f is an isomorphism of the ordered additive group of T onto its multiplicative group of positive elements, with inverse g → log g. As an ordered exponential field, T turns out to be an elementary extension of R exp [START_REF] Van Den Dries | Logarithmic-exponential power series[END_REF].

Transseries can be differentiated termwise; for instance,

n n! e x x n+1
′ = e x x . We obtain a derivation f → f ′ on the field T with constant field {f ∈ T :

f ′ = 0} = R and satisfying (exp f ) ′ = f ′ exp f and (log g) ′ = g ′ /g for f, g ∈ T, g > 0.
Moreover, each f ∈ T has an antiderivative in T, that is, f = g ′ for some g ∈ T.

As in Hardy fields, f > R ⇒ f ′ > 0, for transseries f . We also have a dominance relation on T: for f, g ∈ T we set

f ≼ g :⇐⇒ |f | ⩽ c|g| for some c ∈ R >0 ⇐⇒ (leading transmonomial of f ) ≼ (leading transmonomial of g),
and as in Hardy fields we declare f ≍ g :⇐⇒ f ≼ g and g ≼ f , as well as f ≺ g :⇐⇒ f ≼ g and g ̸ ≼ f . As in Hardy fields we can also differentiate and integrate asymptotic relations: for 0

̸ = f, g ̸ ≍ 1 in T we have f ≼ g iff f ′ ≼ g ′ .
Hardy's ordered exponential field of (germs of) logarithmic-exponential functions embeds uniquely into T so as to preserve real constants and to send the germ x to the transseries x; this embedding also preserves the derivation. However, the field of LE-series enjoys many closure properties that the field of LE-functions lacks. For instance, T is not only closed under exponentiation and integration, but also comes with a natural operation of composition: for f, g ∈ T with g > R we can substitute g for

x in f = f (x) to obtain f • g = f (g(x)). The Chain Rule holds: (f • g) ′ = (f ′ • g) • g ′ . Every g > R has a compositional inverse in T: a transseries f > R with f • g = g • f = x.
As shown in [START_REF] Van Der Hoeven | Transseries and Real Differential Algebra[END_REF], a Newton diagram method can be used to solve any "feasible" algebraic differential equation in T (where the meaning of feasible can be made explicit).

Thus it is not surprising that soon after the introduction of T the idea emerged that it should play the role of a universal domain (akin to Weil's use of this term in algebraic geometry) for asymptotic differential algebra: that it is truly the algebrafrom-which-one-can-never-exit and that it marks an almost impassable horizon for "ordered analysis" [32, p. 148]. Model theory provides a language to make such an intuition precise, as we explain in our survey [START_REF] Aschenbrenner | Toward a model theory for transseries[END_REF] where we sketched a program to establish the basic model-theoretic properties of T, carried out in [START_REF] Aschenbrenner | Asymptotic Differential Algebra and Model Theory of Transseries[END_REF]. Next we briefly discuss our main results from [START_REF] Aschenbrenner | Asymptotic Differential Algebra and Model Theory of Transseries[END_REF].

H-Fields

We shall consider T as an L-structure where the language L has the primitives 0, 1, +, -, • , ∂ (derivation), ⩽ (ordering), ≼ (dominance). More generally, let K be any ordered differential field with constant field Examples include all Hardy fields that contain R, and all ordered differential subfields of T that contain R. In particular, T is an H-field, but T has further basic elementary properties that do not follow from this: its derivation is small, and it is Liouville closed. An H-field K is said to have small derivation if it satisfies f ≺ 1 ⇒ f ′ ≺ 1, and to be Liouville closed if it is real closed and for every f ∈ K there are g, h ∈ K, h ̸ = 0, such that g ′ = f and h ′ = hf . Each Hardy field H has small derivation, and Li(H) is Liouville closed.

C = {f ∈ K : f ′ = 0}.
Inspired by the familiar characterization of real closed ordered fields via the intermediate value property for one-variable polynomial functions, we say that an H-field K has the Intermediate Value Property (IVP) if for all differential polynomials P (Y ) ∈ K{Y } and all f < g in K with P (f ) < 0 < P (g) there is some y ∈ K with f < y < g and P (y) = 0. Van der Hoeven showed that a certain variant of T, namely its H-subfield of gridbased transseries, has IVP; see [START_REF] Van Der Hoeven | A differential intermediate value theorem[END_REF].

Theorem 2.2. The L-theory of T is completely axiomatized by the requirements: being an H-field with small derivation; being Liouville closed; and having IVP.

Actually, IVP is a bit of an afterthought: in [START_REF] Aschenbrenner | Asymptotic Differential Algebra and Model Theory of Transseries[END_REF] we use other (but equivalent) axioms that will be detailed below. We mention the above variant for expository reasons and since it explains why Conjecture A from Section 1 yields that all maximal Hardy fields are elementarily equivalent. Let us define an H-closed field to be an H-field that is Liouville closed and has the IVP. All H-fields embed into H-closed fields, and the latter are exactly the existentially closed H-fields. Thus: Theorem 2.3. The theory of H-closed fields is model complete.

Here is an unexpected byproduct of our proof of this theorem: Corollary 2.4. H-closed fields have no proper differentially algebraic H-field extensions with the same constant field.

IVP refers to the ordering, but the valuation given by ≼ is more robust and more useful. IVP comes from two more fundamental properties: ω-freeness and newtonianity (a differential version of henselianity). These concepts make sense for any differential field with a suitable dominance relation ≼ in which the equivalence

f ≼ g ⇐⇒ f ′ ≼ g ′ holds for 0 ̸ = f, g ≺ 1.
To give an inkling of these somewhat technical notions, let K be an H-field and assume that for every ϕ ∈ K × for which the derivation ϕ∂ is small (that is, ϕ∂O ⊆ O), there exists ϕ 1 ≺ ϕ in K × such that ϕ 1 ∂ is small. (This assumption is satisfied for Liouville closed H-fields.) Let P (Y ) ∈ K{Y } ̸ = . We wish to understand how the function y → P (y) behaves for y ≼ 1. It turns out that this function only reveals its true colors after rewriting P in terms of a derivation ϕ∂ with suitable ϕ ∈ K × .

Indeed, this rewritten P has the form a • (N + R) with a ∈ K × and where N (Y ) ∈ C{Y } ̸ = is independent of ϕ for sufficiently small ϕ ∈ K × with respect to ≼, subject to ϕ∂ being small, and where the coefficients of R(Y ) are infinitesimal. We call N the Newton polynomial of P . Now K is said to be ω-free if for all P as above its Newton polynomial has the form A(Y ) • (Y ′ ) n for some A ∈ C[Y ] and some n. We say that K is newtonian if for all P as above with N (P ) of degree 1 we have P (y) = 0 for some y ∈ O. For H-fields, IVP =⇒ ω-free and newtonian; for Liouville closed H-fields, the converse also holds.

Our main result in [START_REF] Aschenbrenner | Asymptotic Differential Algebra and Model Theory of Transseries[END_REF] refines Theorem 2.3 by giving quantifier elimination for the theory of H-closed fields in the language L above augmented by an additional unary function symbol ι and two extra unary predicates Λ and Ω. These have defining axioms in terms of the other primitives. Their interpretations in T are as follows: ι(f ) = 1/f if f ̸ = 0, ι(0) = 0, and with ℓ 0 := x, ℓ n+1 := log ℓ n ,

Λ(f ) ⇐⇒ f < λ n := 1 ℓ0 + 1 ℓ0ℓ1 + • • • + 1 ℓ0•••ℓn for some n, Ω(f ) ⇐⇒ f < ω n := 1 (ℓ0) 2 + 1 (ℓ0ℓ1) 2 + • • • + 1 (ℓ0•••ℓn) 2 for some n.
Thus Λ and Ω define downward closed subsets of T. The sequence (ω n ) also appears in classical non-oscillation theorems for second-order linear differential equations.

The ω-freeness of T reflects the fact that (ω n ) has no pseudolimit in the valued field T. Here are some applications of this quantifier elimination:

Corollary 2.5.

(1) "O-minimality at infinity": if S ⊆ T is definable, then for some f ∈ T we either have g ∈ S for all g > f in T or g / ∈ S for all g > f in T.

(2) All subsets of R n definable in T are semialgebraic.

Corollaries 2.4 and 2.5 are the departure point for developing a notion of (differential-algebraic) dimension for definable sets in T; see [START_REF] Aschenbrenner | Dimension in the realm of transseries[END_REF].

The results reported on above make us confident that the category of H-fields is the right setting for asymptotic differential algebra. To solidify this impression we return to the motivating examples-Hardy fields, ordered differential fields of transseries, and surreal numbers-and consider how they are related. We start with Hardy fields, which historically came first.

H-Field Elements as Germs

After Theorem 1.2 and Boshernitzan [START_REF] Boshernitzan | New "orders of infinity[END_REF][START_REF] Boshernitzan | Second order differential equations over Hardy fields[END_REF], the first substantial "Hardy field" result on more general differential equations was obtained by van der Hoeven [START_REF] Van Der Hoeven | Transserial Hardy fields[END_REF]. In what follows we use "d-algebraic" to mean "differentially algebraic" and "dtranscendental" to mean "differentially transcendental". These two theorems, when established, imply that all maximal Hardy fields are H-closed. Hence (by Theorem 2.2) they will all be elementarily equivalent to T, and since H-closed fields have the IVP, Conjecture A from Section 1 will follow.

In order to get an even better grasp on the structure of maximal Hardy fields, we also need to understand how to adjoin d-transcendental germs to Hardy fields. An example of this situation is given by d-transcendental series such as n n!!x -n . By an old result by É. Borel [START_REF] Borel | Sur quelques points de la théorie des fonctions[END_REF] every formal power series n a n t n over R is the Taylor series at 0 of a C ∞ -function f on R; then n a n x -n is an asymptotic expansion of the function f (x -1 ) at +∞, and it is easy to show that if this series is d-transcendental, then the germ at +∞ of this function does generate a Hardy field. Here is a far-reaching generalization: Theorem 3.4 (in progress). Every pseudocauchy sequence (y n ) in a Hardy field H has a pseudolimit in some Hardy field extension of H.

The proof of this for H-closed H ⊇ R relies heavily on results from [START_REF] Aschenbrenner | Asymptotic Differential Algebra and Model Theory of Transseries[END_REF], using also intricate glueing techniques. For extensions that increase the value group, we need very different constructions. If succesful, these constructions in combination with Theorem 3.4 will lead to a proof of Conjecture B from Section 1: Theorem 3.5 (in progress). For any countable subsets A < B of a Hardy field H there exists an element y in a Hardy field extension of H with A < y < B.

The case H ⊆ C ∞ , B = ∅ was already dealt with by Sjödin [START_REF] Sjödin | Hardy-fields[END_REF]. The various "theorems in progress" together with results from [START_REF] Aschenbrenner | Asymptotic Differential Algebra and Model Theory of Transseries[END_REF] imply that any maximal Hardy fields H 1 and H 2 are back-and-forth equivalent, which is considerably stronger than H 1 and H 2 being elementarily equivalent. It implies for example Under CH all maximal Hardy fields are isomorphic.

This would be the Hardy field analogue of Corollary 1.1. (In contrast to maximal Hausdorff fields, however, maximal Hardy fields cannot be ℵ 1 -saturated, since their constant field is R.) When we submitted this manuscript, we had finished the proof of Theorem 3.2, and also the proof of Theorem 3.4 in the relevant H-closed case.

Related problems

Some authors (such as [START_REF] Sjödin | Hardy-fields[END_REF]) prefer to consider only Hardy fields contained in C ∞ . Theorem 3.2 and our partial result for Theorem 3.4 go through in the C ∞ -setting. All the above "theorems in progress" are plausible in that setting.

What about real analytic Hardy fields (Hardy fields contained in the subring C ω of C consisting of all real analytic germs)? In that setting Theorem 3.2 goes through. Any d-algebraic Hardy field extension of a real analytic Hardy field is itself real analytic, and so Theorem 3.3 (in progress) will hold in that setting as well. However, our glueing technique employed in the proof of Theorem 3.4 doesn't work there.

Kneser [START_REF] Kneser | Reelle analytische Lösungen der Gleichung ϕ(ϕ(x)) = e x und verwandter Funktionalgleichungen[END_REF] obtained a real analytic solution E at infinity to the functional equation E(x + 1) = exp E(x). It grows faster than any finite iteration of the exponential function, and generates a Hardy field. See Boshernitzan [START_REF] Boshernitzan | Hardy fields and existence of transexponential functions[END_REF] for results of this kind, and a proof that Theorem 3.5 holds for B = ∅ in the real analytic setting. So in this context we also have an abundant supply of Hardy fields.

Similar issues arise for germs of quasi-analytic and "cohesive" functions [START_REF] Écalle | Introduction aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac[END_REF]. These classes of functions are somewhat more flexible than the class of real analytic functions. For instance, the series x -1 + e -x + e -e x + • • • converges uniformly for x > 1 to a cohesive function that is not real analytic.

Accelero-summation

The definition of a Hardy field ensures that the differential field operations never introduce oscillatory behavior. Does this behavior persist for operations such as composition or various integral transforms? In this connection we note that the Hardy field H( R) associated to an o-minimal expansion R of the field of reals is always closed under composition (see [START_REF] Miller | Basics of o-minimality and Hardy fields[END_REF]).

To illustrate the problem with composition, let α be a real number > 1 and let y 0 ∈ C 2 be a solution to (1.2). Then z 0 := y 0 (αx) satisfies the equation

α -2 z ′′ + z = e α 2 x 2 . (3.1)
It can be shown that {y 0 + sin x, z 0 } generates a Hardy field, but it is clear that no Hardy field containing both y 0 + sin x and z 0 can be closed under composition. Adjoining solutions to (1.2) and (3.1) "one by one" as in the proof of Theorem 3.1 will not prevent the resulting Hardy fields to contain both y 0 + sin x and z 0 . In order to obtain closure under composition we therefore need an alternative device. Écalle's theory of accelero-summation [START_REF] Écalle | Introduction aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac[END_REF] is much more than that. Vastly extending Borel's summation method for divergent series [START_REF] Borel | Mémoire sur les séries divergentes[END_REF], it associates to each accelero-summable transseries an analyzable function. In this way many non-oscillating real-valued functions that arise naturally (e.g., as solutions of algebraic differential equations) can be represented faithfully by transseries. This leads us to conjecture an improvement on Theorem 3.1: Conjecture 3.6. Consider the real accelero-summation process where we systematically use the organic average whenever we encounter singularities on the positive real axis. This yields a composition-preserving H-field isomorphism from T da onto a Hardy field contained in C ω .

There is little doubt that this holds. The main difficulty here is that a full proof will involve many tools forged by Écalle in connection with accelero-summation, such as resurgent functions, well-behaved averages, cohesive functions, etc., with some of these tools requiring further elaboration; see also [START_REF] Costin | Asymptotics and Borel Summability[END_REF][START_REF] Menous | Les bonnes moyennes uniformisantes et leur application à la resommation réelle[END_REF].

The current theory of accelero-summation only sums transseries with coefficients in R. Thus it is not clear how to generalize Conjecture 3.6 in the direction of Theorem 3.3. Such a generalization might require introducing transseries over a Hardy field H with suitable additional structure, as well as a corresponding theory of accelero-summation over H for such transseries. In particular, elements of H should be accelero-summable over H in this theory, by construction.

H-Field Elements as Generalized Transseries

Next we discuss when H-fields embed into differential fields of formal series. A classical embedding theorem of this type is due to Krull [START_REF] Krull | Allgemeine Bewertungstheorie[END_REF]: any valued field has a spherically complete immediate extension. As a consequence, any real closed field containing R is isomorphic over R to a subfield of a Hahn field R[[M]] with divisible monomial group M, such that the subfield contains R(M). We recently proved an analogue of this theorem for valued differential fields [START_REF] Aschenbrenner | Maximal immediate extensions of valued differential fields[END_REF]. Here a valued differential field is a valued field of equicharacteristic zero equipped with a derivation that is continuous with respect to the valuation topology. For a real closed H-field K with constant field C this theorem gives a Hahn field

K = C[[M]
] with a derivation ∂ on K making it an H-field with constant field C such that K is isomorphic over C to an H-subfield of K that contains C(M). A shortcoming of this result is that there is no guarantee that ∂ preserves infinite summation. In contrast, the derivation of T is strong (does preserve infinite summation). An abstract framework for even more general notions of transseries is due to van der Hoeven and his former student Schmeling [START_REF] Schmeling | Corps de transséries[END_REF].

Fields of transseries

To explain this, consider an (ordered) Hahn field R[[M]] with a partially defined function exp obeying the usual rules of exponentiation; see [52, Section 4.1] for details. In particular, exp has a partially defined inverse function log. We say that R[[M]] is a field of transseries if the following conditions hold:

(T1) the domain of the function log is R[[M]] >0 ; (T2) for each m ∈ M and n ∈ supp log m we have n ≻ 1; (T3) log(1 + ε) = ε -1 2 ε 2 + 1 3 ε 3 + • • • for all ε ≺ 1 in R[[M]]
; and (T4) for every sequence (m n ) in M with m n+1 ∈ supp log m n for all n, there exists an index n 0 such that for all n ⩾ n 0 and all n ∈ supp log m n , we have n ≽ m n+1 and (log m n ) mn+1 = ±1.

The first three axioms record basic facts from the standard construction of transseries. The fourth axiom is more intricate and puts limits on the kind of "nested transseries" that are allowed. Nested transseries such as

y = √ x + e √ log x+e √ log log x+e ••• (4.1)
are naturally encountered as solutions of functional equations, in this case

y(x) = √ x + e y(log x) . (4.2)
Axiom (T4) does allow nested transseries as in (4.1), but excludes series like

u = √ x + e √ log x+e √ log log x+e ••• + log log log x + log log x + log x,
which solves the functional equation u(x) = √

x + e u(log x) + log x; in some sense, u is a perturbation of the solution y in (4.1) to the equation (4.2).

Schmeling's thesis [START_REF] Schmeling | Corps de transséries[END_REF] shows how to extend a given field of transseries K = R[[M]] with new exponentials and nested transseries like (4.1), and if K also comes with a strong derivation, how to extend this derivation as well. Again, (T4) is crucial for this task: naive termwise differentiation leads to a huge infinite sum that turns out to be summable by (T4). A transserial derivation is a strong derivation on K such that nested transseries are differentiated in this way. Such a transserial derivation is uniquely determined by its values on the log-atomic elements: those λ ∈ K for which λ, log λ, log log λ, . . . are all transmonomials in M.

We can now state a transserial analogue of Krull's theorem. This analogue is a consequence of Theorem 5.3 below, proved in [START_REF] Aschenbrenner | The surreal numbers as a universal H-field[END_REF].

Theorem 4.2. Every H-field with small derivation and constant field R can be embedded over R into a field of transseries with transserial derivation.

For simplicity, we restricted ourselves to transseries over R. The theory naturally generalizes to transseries over ordered exponential fields [START_REF] Van Der Hoeven | Transseries and Real Differential Algebra[END_REF][START_REF] Schmeling | Corps de transséries[END_REF] and it should be possible to extend Theorem 4.2 likewise.

Hyperseries

Besides derivations, one can also define a notion of composition for generalized transseries [49,[START_REF] Schmeling | Corps de transséries[END_REF]. Whereas certain functional equations such as (4.2) can still be solved using nested transseries, solving the equation E(x + 1) = exp E(x) where E(x) is the unknown, requires extending T to a field of transseries with composition containing an element E(x) = exp ω x > T, called the iterator of exp x. Its compositional inverse log ω x should then satisfy log ω log x = (log ω x)-1, providing us with a primitive for (x log

x log 2 x • • • ) -1 : log ω x = dx x log x log 2 x • • • .
It is convenient to start with iterated logarithms rather than iterated exponentials, and to introduce transfinite iterators log α x recursively using log α x = dx β<α log β x (α any ordinal).

By Écalle [START_REF] Écalle | Introduction aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac[END_REF] the iterators log α x with α < ω ω and their compositional inverses exp α x suffice to resolve all pure composition equations of the form

f •k1 • ϕ 1 • • • • • f •kn • ϕ n = x where ϕ 1 , . . . , ϕ n ∈ T and k 1 , . . . , k n ∈ N.
The resolution of more complicated functional equations involving differentiation and composition requires the introduction of fields of hyperseries: besides exponentials and logarithms, hyperseries are allowed to contain iterators exp α x and log α x of any strength α. For α < ω ω , the necessary constructions were carried out in [START_REF] Schmeling | Corps de transséries[END_REF]. The ultimate objective is to construct a field Hy of hyperseries as a proper class, similar to the field of surreal numbers, endow it with its canonical derivation and composition, and establish the following: Conjecture 4.3. Let Φ be any partial function from Hy into itself, constructed from elements in Hy, using the field operations, differentiation and composition. Let f < g be hyperseries in Hy such that Φ is defined on the closed interval [f, g] and Φ(f )Φ(g) < 0. Then for some y ∈ Hy we have Φ(y) = 0 and f < y < g. One might then also consider H-fields with an additional composition operator and try to prove that these structures can always be embedded into Hy.

Growth Rates as Numbers

Turning to surreal numbers, how do they fit into asymptotic differential algebra?

The H-field of surreal numbers

a + b := a L + b, a + b L a R + b, a + b R ab := a L b + ab L -a L b L , a R b + ab R -a R b R a L b + ab R -a L b R , a R b + ab L -a R b L .
Remarkably, these operations make No into a real closed field with < as its field ordering and with R uniquely embedded as an initial subfield. (A set A ⊆ No is said to be initial if for all a ∈ A all b < s a are also in A.)

Can we use such magical recursions to introduce other reasonable operations? Exponentiation was dealt with by Gonshor [START_REF] Gonshor | An Introduction to the Theory of Surreal Numbers[END_REF]. But it remained long open how to define a "good" derivation ∂ on No such that ∂(ω) = 1. (An ordinal α is identified with the surreal of length α whose sign sequence has just plus signs.) A positive answer was given recently by Berarducci and Mantova [START_REF] Berarducci | Surreal numbers, derivations, and transseries[END_REF]. Their construction goes in two parts. They first analyze No as an exponential field, and show that it is basically a field of transseries in the sense of Section 4. A transserial derivation on No is determined by its values at log-atomic elements. There is some flexibility here, but [START_REF] Berarducci | Surreal numbers, derivations, and transseries[END_REF] presents a "simplest" way to choose these derivatives. Most important, that choice indeed leads to a derivation ∂ BM on No. In addition: Theorem 5.1 (Berarducci-Mantova [START_REF] Berarducci | Surreal numbers, derivations, and transseries[END_REF]). The derivation ∂ BM is transserial and makes No a Liouville closed H-field with constant field R.

This result was further strengthened in [START_REF] Aschenbrenner | The surreal numbers as a universal H-field[END_REF], using key results from [START_REF] Aschenbrenner | Asymptotic Differential Algebra and Model Theory of Transseries[END_REF]: Theorem 5.2. No with the derivation ∂ BM is an H-closed field.

Embedding H-fields into No

In the remainder of this section we consider No as equipped with the derivation ∂ BM , although Theorems 5.1 and 5.2 and much of what follows hold for other transserial derivations. Returning to our main topic of embedding H-fields into specific H-fields such as No, we also proved the following in [START_REF] Aschenbrenner | The surreal numbers as a universal H-field[END_REF]: Theorem 5.3. Every H-field with small derivation and constant field R can be embedded as an ordered differential field into No.

How "nice" can we take the embeddings in Theorem 5.3? For instance, when can we arrange the image of the embedding to be initial? The image of the natural embedding T → No is indeed initial, as has been shown by Elliot Kaplan.

For further discussion it is convenient to introduce, given an ordinal α, the set No(α) := a ∈ No : ℓ(a) < α . It turns out that for uncountable cardinals κ, No(κ) is closed under the differential field operations, and in [START_REF] Aschenbrenner | The surreal numbers as a universal H-field[END_REF] we also show: In particular, the H-field No(ω 1 ) of surreal numbers of countable length is an elementary submodel of No. It has the η 1 -property: for any countable subsets A < B of No(ω 1 ) there exists y ∈ No(ω 1 ) with A < y < B. This fact and the various "theorems in progress" from Section 3 imply:

Under CH all maximal Hardy fields are isomorphic to No(ω 1 ). This would be an analogue of Ehrlich's observation about maximal Hausdorff fields.

Hyperseries as numbers and vice versa

The similarities in the constructions of the field of hyperseries Hy and the field of surreal numbers No led van der Hoeven [52, p. 6] to the following: Conjecture 5.5. There is a natural isomorphism between Hy and No that associates to any hyperseries f (x) ∈ Hy its value f (ω) ∈ No.

The problem is to make sense of the value of a hyperseries at ω. Thanks to Gonshor's exponential function, it is clear how to evaluate ordinary transseries at ω. The difficulties start as soon as we wish to represent surreal numbers that are not of the form f (ω) with f (x) an ordinary transseries. That is where the iterators exp ω and log ω come into play: The intuition behind Conjecture 5.5 is that all "holes in No can be filled" using suitable nested hyperseries and suitable iterators of exp and log. It reconciles two a priori very different types of infinities: on the one hand, we have growth orders corresponding to smooth functional behavior; on the other side, we have numbers. Being able to switch between functions (more precisely: formal series acting as functions) and numbers, we may also transport any available structure (For a family (a i ) of positive surreals, i a i := exp i log a i if i log a i is defined.) Since ∂ BM is transserial, it does admit a distinguished right inverse ∂ -1 BM . According to [START_REF] Berarducci | Surreal numbers, derivations, and transseries[END_REF]Remark 6.8], ∂ BM λ = 1/ log ′ ω λ for log-atomic λ with λ > exp n ω for all n. For λ = exp ω ω and setting exp ′ ω (a) := n log n exp ω a for a ∈ No >0 , this yields ∂ BM λ = exp ′ ω ω, which is also the value we expect for ∂ c λ. However, for λ = exp ω (exp ω ω) we get ∂ BM λ = exp ′ ω (exp ω ω) whereas we expect ∂ c λ = (exp ′ ω ω) • exp ′ ω (exp ω ω). Thus the "simplest" derivation ∂ BM making No an H-field probably does not coincide with the ultimately "correct" derivation ∂ c on No. Berarducci and Mantova [START_REF] Berarducci | Transseries as germs of surreal functions[END_REF] use similar considerations to conclude that ∂ BM is incompatible with any reasonable notion of composition for surreal numbers.

The surreal numbers from a model theoretic perspective

We conclude with speculations motivated by the fact that various operations defined by "surreal" recursions have a nice model theory. Each of these theories is model complete in a natural language. Is there a model theoretic reason that explains why this works so well?

Let us look at this in connection with the last example. Our aim is to define a derivation ∂ on No making it an H-field. Let a ∈ No be given for which we wish to define ∂a, and assume that ∂b has been defined for all b ∈ L a ∪ R a . Let ∆ a be the class of all surreals b for which there exists a derivation ∂ on No with ∂a = b and taking the prescribed values on L a ∪R a . Assembling all conditions that should be satisfied by ∂a, it is not hard to see that there exist sets L, R ⊆ No such that ∆ a = {b ∈ No : L < b < R}. We are left with two main questions: When do we have L < R, thereby allowing us to define ∂a = {L | R}? Does this lead to a global definition of ∂ on No making it an H-closed field? It might be of interest to isolate reasonable model theoretic conditions that imply the success of this type of construction. If the above construction does work, yet another question is whether the resulting derivation coincides with ∂ BM .

  This yields a dominance relation ≼ on K by f ≼ g :⇐⇒ |f | ⩽ c|g| for some positive c ∈ C and we view K accordingly as an L-structure. The convex hull of C in K is the valuation ring O = {f ∈ K : f ≼ 1} of K, with its maximal ideal O := {f ∈ K : f ≺ 1} of infinitesimals. Definition 2.1. An H-field is an ordered differential field K such that (with the notations above), O = C + O, and for all f ∈ K we have: f > C ⇒ f ′ > 0.

Theorem 3 . 1 .

 31 The differential subfield T da of T whose elements are the d-algebraic transseries is isomorphic over R to a Hardy field.The proof of this theorem is in the spirit of model theory, iteratively extending by a single d-algebraic transseries. The most difficult case (immediate extensions) is handled through careful construction of suitable solutions as convergent series of iterated integrals. We are currently trying to generalize Theorem 3.1 to d-algebraic extensions of arbitrary Hardy fields. Here is our plan: Theorem 3.2. Every Hardy field has an ω-free Hardy field extension. Theorem 3.3 (in progress). Every ω-free Hardy field has a newtonian d-algebraic Hardy field extension.

Theorem 4 . 1 .

 41 Every valued differential field has a spherically complete immediate extension.

Theorem 5 . 4 .

 54 The H-subfield No(κ) of No is an elementary submodel of No.

2 := ω 2 ,

 22 exp ω ω := {ω, exp ω, exp 2 ω, . . . | } log ω ω := {R | . . . , log 2 ω, log ω, ω} exp 1/2 ω := exp ω log ω ω + 1 exp log 2 ω, exp 2 log 2 2 ω, . . . . . . , exp 2 log ω, exp √ ω

1 n

 1 in both directions: we immediately obtain a canonical derivation ∂ c (with constant field R) and composition • c on No, as well as a notion of simplicity on Hy. Does the derivation ∂ BM coincide with the canonical derivation ∂ c induced by the conjectured isomorphism? A key observation is that any derivation ∂ on No with a distinguished right inverse ∂ -1 naturally gives rise to a definition of log ω : log ω a := ∂ -1 (∂a log ′ ω a) where log ′ ω a := log n a (a ∈ No, a > R).

  Examples: (No; ⩽, +, • ) is a model of the theory of real closed fields; (No; ⩽, +, • , exp) is a model of the theory of R exp ; and (No; ⩽, +, • , ∂ BM ) is a model of the theory of H-closed fields.

  The totality No of surreal numbers is not a set but a proper class: a surreal a ∈ No is uniquely represented by a transfinite sign sequence (a λ ) λ<ℓ(a) ∈ {-, +} ℓ(a) where ℓ(a) is an ordinal, called the length of a; a surreal b is said to be simpler than a (notation: b < s a) if the sign sequence of b is a proper initial segment of that of a. Besides the (partial) ordering < s , No also carries a natural (total) lexicographic ordering <. For any sets L < R of surreals there is a unique simplest surreal a with L < a < R; this a is denoted by {L | R} and called the simplest or earliest surreal between L and R. In particular, a = {L a | R a } for any a ∈ No, where L a := {b < s a : b < a} and R a = {b < s a : b > a}. We let a L range over elements of L a , and a R over elements of R a .A rather magical property of surreal numbers is that various operations have natural inductive definitions. For instance, we have ring operations given by

The first-named author was partially support by NSF Grant DMS-1700439.


Tarek Ameid
email: tarek.ameid@ieee.org

Hicham Talhaoui
email: hichem.talhaoui@univ-bba.dz

Abdelkarim Ammar
email: a.ammar@unive-boumerdes.dz

Younes Azzoug
email: younes.azzoug@univ-biskra.dz

Remus Pusca
email: puscaremus@hotmail.com

Raphaël Romary
email: raphael.romary@univ-artois.fr

Jean-Philippe Lecointe
email: jphilippe.lecointe@univ-artois.fr

Hardware Implementation of Modified Backstepping Control for Sensorless Induction Motor Drive

Keywords: Induction motor, Backstepping control, Luenberger observer, dSpace 1104. I

This paper presents a hardware implementation of sensorless Backstepping control for induction motor drive. The presented Backstepping control scheme has been designed in the stationary reference frame to reduce the control algorithm complexity. Furthermore, a full order Luenberger observer has been proposed for speed and torque estimation. Several operation conditions of the IM have been conducted such as load application, speed direction reversal and low-speed condition. In addition, an industrial benchmark test which comprises all speed condition has been done to check the control ability in different operation points. This work presents for the first time the experimental implementation of this developed control algorithm. The obtained experimental results prove the effectiveness and performance of the proposed control scheme.

INTRODUCTION

The objectives of the various industrial structures are always linked to increasing quality, productivity and profitability. Three-phase squirrel-cage Induction Motors (IMs) are widely used in variable speed drives, particularly in high-tech domains due to of their high power-to-weight ratio, robust structure, low purchase cost, and easy maintenance [START_REF] Sahoo | Field weakening strategy for a vector-controlled induction motor drive near the six-step mode of operation[END_REF]. However, the difficulty of using IM in control-loop lies in the fact of the nonlinear coupled mathematical model, multivariable, which involves parameters that may vary with temperature, frequency, and other operating conditions. These constraints have significant effects on the accuracy of speed and torque control.

Many different methods are available for controlling IMs, which are distinguished by the motor performance they offer, but also by the cost of implementation. The Field-Oriented Control (FOC) has led to a radical change in control of the IMs. In this strategy, the torque and flux are decoupled by a suitable decoupling network and controlled independently by quadratic and direct stator currents respectively as a separately excited DC motor [START_REF] Casadei | FOC and DTC: two viable schemes for induction motors torque control[END_REF]. Unfortunately, the use of PI controllers needs particular information on control system modeling. Moreover, under these conditions, PI controllers have limitation provoked by the presence of time-varying parameters, disturbances, and uncertainties which affect the stability and the dynamic of the system [START_REF] Ammar | Performance improvement of direct torque control for induction motor drive via fuzzy logic-feedback linearization[END_REF]. Therefore, when the controlled part is subjected to strong non-linearities and time variables, the control algorithms must be designed to ensure the robustness of the process behavior with regard to uncertainties on the parameters and their variations.

The successful application of mathematical tools achieved a breakthrough development in the so-called robust and nonlinear control approaches to solving this problem and has made great progress recently. Non-linear controls offer several advantages like the good dynamic and the highperformance over linear control schemes. Various robust and nonlinear control strategies for IM drives have been proposed in the literature, like Sliding Mode Control (SMC) [START_REF] Benchaib | Sliding mode input-output linearization and field orientation for real-time control of induction motors[END_REF], the Input-Output Feedback Linearization control (IOFL) [START_REF] Alonge | Input-output feedback linearization control with on-line MRAS-based inductor resistance estimation of linear induction motors including the dynamic end effects[END_REF] and Backstepping Control (BC) [START_REF] Echeikh | High performance backstepping control of a fivephase induction motor drive[END_REF]. The SMC adjustment is basically a method that forces the response to slide along a predefined trajectory. However, the chattering phenomena are the major disadvantage of this method [START_REF] Guezmil | Experimental Investigation of Passive Fault Tolerant Control for Induction Machine Using Sliding Mode Approach[END_REF]. Moreover, the IOFL provides an exact decoupling between system variables which preserves good behavior in dynamic states. However, the stability of this strategy cannot be guaranteed due to the presence of system uncertainties [START_REF] Lascu | Direct Torque Control With Feedback Linearization for Induction Motor Drives[END_REF]. The BC is based on the tools of Lyapunov's stability, and offers a great flexibility in the syntheses. It has shown a fast dynamic response, easy implementation and robustness to parameters variation, and also for load torque disturbances [START_REF] Moutchou | Sensorless speed backstepping control of induction machine, based on speed MRAS observer[END_REF]. The comparative study between integral and classical Backstepping in indirect field orientation based-control structures of the IM is presented in [START_REF] Fateh | Comparative study of integral and classical backstepping controllers in IFOC of induction motor fed by voltage source inverter[END_REF]. In [START_REF] Trabelsi | Backstepping control for an induction motor using an adaptive sliding rotor-flux observer[END_REF] the rotor flux is recovered by using an adaptive sliding mode observer for the Backstepping driven IM. A field-programmable gate array (FPGA)-based implementation for an adaptive Backstepping control with radial basis function network (RBFN) observer is proposed in [START_REF] Teng | FPGA-based adaptive backstepping control system using RBFN for linear induction motor drive[END_REF].

The control techniques cannot guarantee a good performance without the use of a suitable state observer. Many observer's structures have been associated with Backstepping control for different parameters estimation in order to minimize the cost of the control system by reducing the number of sensors. In [START_REF] Zaafouri | DSP-based adaptive backstepping using the tracking errors for high-performance sensorless speed control of induction motor drive[END_REF], the Backstepping observer for the rotor speed estimation is presented. The authors in [START_REF] Ameid | Sensorless speed estimation and backstepping control of induction motor drive using model reference adaptive system[END_REF] propose a sensorless Backstepping control for the IM drive based on model reference adaptive system (MRAS) for rotor speed and rotor flux estimations.

Usually, the design of the Backstepping control is developed in the synchronous (d-q) reference frame in most works. It can be seen that the combination of Backstepping strategy with rotor field-orientation principle provides good control performance. However, the field-oriented reference frame requires the coordinate transformation which increases the complexity of the control's algorithm. Moreover, the estimation of flux position which is necessary for the transformation has some difficulties, particularly in the experimental implementation. Hence, to overcome this shortcoming, a proposed control scheme which developed in the stationary reference frame has been presented in [START_REF] Benzineb | A PI / Backstepping Approach for Induction Motor Drives Robust Control[END_REF], [START_REF] Moutchou | MRAS-based sensorless speed backstepping control for induction machine, using a flux sliding mode observer[END_REF]. In [START_REF] Benzineb | A PI / Backstepping Approach for Induction Motor Drives Robust Control[END_REF], a combined control law between the traditional PI controllers and the Backstepping strategy has been presented, while an MRAS and sliding mode observers have been used for speed and flux component estimations for Backstepping-driven IM in [START_REF] Moutchou | MRAS-based sensorless speed backstepping control for induction machine, using a flux sliding mode observer[END_REF]. However, in these works, only the simulation has been performed. This paper is devoted to the nonlinear control of the induction machine. Mathematical modeling and experimental validation are investigated. The Backstepping control strategy is designed in the stationary reference frame in order to reduce the control algorithm complexity with keeping the same good propertied of the conventional Backstepping control. Moreover, a full order Luenberger observer has been used for rotor speed and torque estimations; investigate a sensorless control algorithm by decreasing the cost and improve the reliability of drive system. The proposed sensorless control is examined under different speed region operations. The obtained results show high robustness and good estimation accuracy in several modes of operation.

II. MODEL OF THE IM

The dynamic equations of IM can be expressed as follows:

Bu Ax x    (1) with:     T r r s s i i x        ] [ ] [ s  u u u s                                     r r r r r r T T M T T M T T A 1 0 1 0 0 0       ;                0 0 0 0 0 0 B where:                     1 1 1 1 1 1 s r s L M T T         , r s L L M 2 1   , s s s R L T  , r r r R L T 
The mechanical equations must also consider:

                   ) ( r e s s s s s e C C f dt d J i i L M P C     (2) with: P   

III. BACKSTEPPING CONTROL OF IM

A. First step

In the first step, the rotor speed and rotor flux module tracking error are defined by:

             z 2 2 1 1 r ref ref e (3)
where

2 2   r r r      (4)
Also, the error dynamic equations can be written by:

                               T 2 ) ( T M 2 - z ) ( J r 2 r 2 1 1 r s r s r ref r s r s r ref i i dt d J C J f i i e             (5)
with:

s L M P    By putting the virtual control expressions below,                      s r s r s r s r i i i i 1 1 (6)
We can write:

                         T 2 T M 2 - z J r 2 1 r 2 1 1 1 r ref r ref dt d J C J f e       (7)
Let us check the tracking error stability by choosing the Lyapunov Candidate Function (LCF) below:

2 2 2 1 1 2 1 2 1 e e    (8)
The derivative of this equation gives:

2 2 1 1 1 e e e e       (9)
Let us consider the following virtual control reference that stabilizes the tracking errors e1 and z1:

                                  T 2 z d 2 r 2 2 1 1 1 1 1 1ref r ref r ref r ref dt d M T J C J f e c J     (10)
where c1 > 0 and d1 > 0 are the positive design gains that determine the dynamic of the closed loop. so:

       z 1 1 1 1 1 1 z d e c e   (11) 0 2 1 1 2 1 1 1      z d e c  ( 12
)
This is evidently semi negative definite, so the tracking errors e1 and z1 have been stabilized.

B. Second step

Previous references, chosen to ensure a stable dynamic of speed-and flux-tracking error, cannot be imposed on the virtual controls without considering errors between them.

Let us define the following errors:

       1 1 2 1 1 2 z     ref ref e (13)
One determines the new dynamics of the errors e1 and z1, expressed now in terms of e2 and z2, by: By deriving e2 and z2, we obtain the following error dynamics equations:

             2 r
                             s r s r s r s r u u z u u e 2 2 2 2   (15)
where:

                                             p i i T M T T T s s r r r r ref r r ref                2 2 2 1 1 1 2 2 1 1 1 2 1 1   , r s L L M    Now that the real control variables (  s u ,  s u ) have
appeared in this equation, we define the final complete LCF as follows: where c2 and d2 are the positive design gains that determine the dynamics of the closed loop. In order to make the LCF derivative be negative definite and have the following expression:

2 2 2 2 2 1 2 1 2 2 1 2 1 2 1 2 1 z e z e      (16) Thus, the LCF derivative is determined below,                                                    s r s r
0 2 2 2 2 2 2 2 1 1 2 1 1 2        z d e c z d e c  (18)
We chose voltage control as follows:

                         0 z T M 2 0 2 1 r 2 2 2 1 2 2              s r s r s r s r u u z d u u e J e c ( 19
)
And then we obtain the input controls, (

 s u ,  s u ):                              r b r b r s r b r b r s A B u A B u 2 2 1 1 (20)
where:

            

IV. SENSORLESS CONTROL ALGORITHM

The Luenberger state observer for estimating the stator current and the rotor flux, using the measured stator currents and voltages, is described by the following set of equations:

          ) ( ) ( ˆ) ( ) ( ˆt x C t y y L Bu t x LC A x  (21) with:     T r r s s i i x        ˆ,     T s s i i y    ,     T s s i i y    ,        0 0 1 0 0 0 0 1 C
where ^ denotes the estimated values, ) t ( x ˆis the observer state vector and L is the observer gain matrix, which is selected so that the system will be stable [START_REF] Chalawane | MRAS and Luenberger observers using a SIFLC controller in adaptive mechanism based sensorless fuzzy logic control of induction motor[END_REF]. The principle of the observer diagrams is presented in Fig. 1. To ensure that the estimation error vanish over time for any, ) 0 ( x ˆwe should select the observer gain matrix L so that (A-LC) is asymptotically stable. Therefore, the observer gain matrix should be chosen so that all Eigen values of (A-LC) have negative real parts.

      ) t ( Cx ) t ( y ) t ( Bu ) t ( Ax ) t ( x          ) t ( x Ĉ ) t ( y ˆ)) t ( y ) t ( y ( L ) t ( Bu ) t ( x  ) t ( x ˆ + - L ) t ( y ) t ( x û ) t ( y ˆ
The gain matrix L is defined as follows:

T L L L L L L L L g g g g g g g g L          3 4 1 2 4 3 2 1
where:

             r r s s g L T T L R k g 1 1 ) 1 ( 1    ;  ) 1 ( 2   g L k g ;                                           r r s L g r L r s L g L T T T a k T a T T a k g 1 1 1 1 1 1 1 2 3       ;  ) 1 ( 4 L g L a k g   .
with:

rc sc L L L M a   and ) 1 (  g k
According to Lyapunov's theory, the adaptive scheme for speed estimation is given by:

dt e e k e e k r isb r is i r isb r is p          ) . . ( ) . . ( ˆ       (22)
where:

x x e   ,
kp and ki are respectively the proportional and integral constants.

V. EXPERIMENTAL IMPLEMENTATION

The experimental test bench of control is shown in Fig. 2. The control algorithm is implemented using dSpace DS1104 R&D controller board with control desk and Matlab/Simulink software package. The test bench is composed of: an IM of 1.1 kW, dSpace DS 1104 with control desk software plugged into the computer, a power electronics SEMIKRON converter with a rectifier and an IGBT inverter, a magnetic powder brake with load control unit, a speed sensor (incremental encoder), Hall's type current and voltage sensors and an autotransformer (0-450V). The specifications for the IM with corresponding parameters are given in appendix.

VI. APPLICATION AND RESULTS

The figures below show the experimental results of the sensorless Backstepping control of the IM drive based on the Luenberger observer. The chosen sampling frequency for the Real-Time implementation is 10kHz.

A. Reference tracking test:

a benchmark reference trajectory is chosen in order to test the robustness of the IM control vis-a-vis to speed tracking undervalues (0, 100, 300, 1200, 0, 50 rpm), as presented in Fig. 3. The test of speed direction reversing has been employed in the figure below (Fig. 4). The speed reference is changing from (±1000 rpm). The Fig. 4a illustrates from the top to the bottom the rotor speed (1div=1000rpm) and the electromagnetic torque (1div=10N.m). Next in Fig. 4b, the stator flux components, flux magnitude and angle position are depicted respectively. As it can be seen from Fig. 4a, the Backstepping control presents good speed response at different operation points and provides a considerable reduction in torque ripples.

Further, in Fig. 4b the stator flux components illustrate a good sinusoid and smooth waveform. Then, the flux magnitude has shown low ripples and fast tracking to its reference's value of 1Wb. Finally, the angel position is presented in the bottom to indicate the rotation sense and frequency. The rotor speed and torque evolution are presented in Fig. 5. The results clearly show that the speed converges rapidly towards its reference without overshooting. The perturbation caused by the application of the load is quickly eliminated. The electromagnetic torque is reacting very quickly to load variations without reaching an unacceptable value. Fig. 6 presents the stator phase currents, it can be seen that the stator phase currents have a good dynamic and balanced sinusoid waveform.

B. Starting up, steady state and load application

ref  and 2 ref

 2 are present respectively the reference of speed and rotor flux modulus.

2

 2

Fig. 1 .

 1 Fig. 1. Principle of the observer diagrams

Fig. 2 .

 2 Fig. 2. Test bench of the Backstepping control.

.

Fig. 3 .

 3 Fig. 3. Speed response according to the reference trajectory

Fig. 4 .

 4 Fig. 4. Speed reversal tests, (a) speed and torque (b) flux components, flux magnitude and angel position.

Fig. 5 .

 5 Fig. 5. Rotor speed and torque with rated load application

Fig. 6 .

 6 Fig. 6. Stator phase currents with rated load application VII. CONCLUSION In this paper, the experimental validation of a sensorless nonlinear Backstepping control for IM drive is performed. The stationary reference frame-based model has been used for the control algorithm design. Furthermore, the full order Luenberger observer has been applied for rotor speed and torque estimation a sensorless control algorithm. The effectiveness of the global control strategy was examined through the experimental implementation. The obtained results show the high performance and the good dynamics of the proposed control algorithm during different operation condition. Moreover, the Luenberger observer offers an accurate speed estimation which improves the reliability of sensorless control. The use of stationary reference framebased model reduces the control algorithm complexity and overcome the problem position estimation while keeping the same good properties of the backstepping control.As a perspective of this paper, the Backstepping control based on stationary reference frame model will be designed for the fault tolerance control of the IM, diagnosis in closedloop drive and efficiency optimization algorithm based on losses minimization.APPENDIXType 3-phase squirrel-cage Rated Power 1,1 kW Number of pole pairs 2 Rated Speed 1450 rpm Rated Stator Current 2,5 A Rated RMS phase voltage 400 V Stator resistance 6.75  Rotor resistance 6.21  Stator inductance 0.5192 H Rotor inductance 0.5192 H Mutual inductance 0.4757 H Inertia moment 0.0124 kg.m 2 damping coefficient 0.0029N.m/rad/s