
In my thesis, I have shown how to represent surreal numbers, as defined
by
Hyperseries are generalized power series in a variable , which are endowed with a hyperserial structure: a collection of functions which are regular (e.g. analytic and monotone) in a formal sense, and which are themselves represented as hyperseries . These functions include
exponentials and logarithms ,
socalled hyperexponentials, e.g. a formal term which satisfies Abel's equation
and its functional inverse with ,
socalled nested series such as expansions with transfinite depth
(1) 
which can be made rigorous sense of in hyperseries.
My thesis consisted in showing that the class of surreal numbers has a natural hyperserial structure. This raised several research questions that I am now interested in, and which I summarize below.
Using the hyperserial structure on the nonArchimedean ordered field extension , it is possible to represent surreal numbers as hypereries with real coefficients. This representation gives a natural way to treat numbers as functions defined on , and to differentiate them as such. In other words, this gives a canonical way to define a derivation and a composition law on surreal numbers. I have notes on how to define these operations, that I plan to turn into a series of papers.
I want to study several problems related to the firstorder structure
This structure has interesting properties: closure under conjugacy equations, model completeness as an ordered valued differential field, setwise order saturation...
Nevertheless, it is difficult to even start studying its firstorder properties when taking the composition law into account, because not much is known about the model theory of ordered structures with (noncommutative) composition laws. I plan to propose interesting firstorder theories accounting for such structures, and to make baby steps toward understanding , which I believe is a prime and natural example of those.
One way to start is restrain ourselves to studying a small part of the language which includes the composition law. Accordingly, the natural candidate is the biordered group . This group shares many firstorder properties with ordered groups of germs (at ) of definable unary functions in ominimal expansions of realclosed fields.
I want to draw on this connection to further the knowledge about these groups, as well as more abstract ordered groups which I call growth order groups. The elementary class of growth order groups in the language of ordered groups is, roughly speaking, characterized by way the inequality
is solved in such groups. I studied in detail [3] a prominent example of growth order group, which is a small subgroup of .
I expect that ominimality is a natural source of growth ordered groups. In particular, I want to prove or disprove the following conjecture:
Conjecture
I can prove this in the particular case when the expansion of with the exponential function is levelled in the sense of [12], and I would like to generalize this.
I expect that the noncommutative valuation theory of a growth orderd group retains certain features of the valuation theory of Abelian ordered groups. Just as valuation theory gives tools to obtain asymptotic expansions of regular growth rates of an additive nature, I expect that growth rates of functions definable in certain ominimal structures can be decomposed as noncommutative compositions of simpler growth rates. Growth order groups are a way to make this idea precise. Another way is to find a formal framework in which transfinite noncommutative product make sense and can be made to form a growth order group. I have an ongoing project of defining such structures, and showing that certain groups of positive infinite transseries are of this form.
A crucial feature of surreal numbers is the possibility of defining operations on Cartesian powers of via recursive definitions as per [9]. Indeed this is how the arithmetic [6] and exponential function [10] and hyperserial structure [4] on were defined. It is sometimes possible [9] to show that a function with recursive definition on is “tame”, for instance satisfying the intermediate value property (IVT). This is particularly interesting because having the IVT for unary terms in a firstorder language is sometimes conducive to proving existential closedness for the corresponding structure (e.g. adding the IVT for terms to linearly ordered Abelian groups, ordered domains, and Liouvilleclosed Hfields with small derivation gives existential closedness).
Thus it would be interesting to generalize previous constructions of opertions on with a more model theoretic approach. In particular given a firstorder language with as the only relation symbol, and an theory of dense linear orders without endpoints, when and how (and in what order) can one define, in a recursive way, interpretations of the function symbols in as functions on and its Cartesian powers, in such a way that be a model of ? When doing so, what is the complete theory which we obtain?
One long standing open question in ominimality is the existence of a transexponential ominimal expansion of the real ordered exponential field, i.e. an expansion which defines a unary function growing faster than any finite iterate of . Abel's equation in
(3.1) 
is the simplest functional equation whose solutions in Hardy fields [5] are transexponential.
On the realanalytic side, Kneser's [11] solution to (3.1) on is a
natural candidate for such ominimal investigations. On the
formalsurreal side, I hope that the calculus of hyperseries on faithfully represents asymptotic properties (at ) of .
Recently, Adele
Together with A.
Conjecture
Conjecture
A unique feature of surreal numbers is that they naturally contain nested numbers, e.g. numbers which expand as
(3.2) 
where is a specific surreal number which plays the role of a positive infinite variable.
On the geometric/analytic side, the functional equation
(3.3) 
naturally generate germs which, when represented using logarithmicoexponential terms, expand in a similar way as (3.2). There are good reasons [2] to believe that the behavior of differential polynomials on these “nested germs” is exactly the same as their behavior on the corresponding nested numbers. Exploiting this, I want to study how the functional equation (3.2) can be solved in Hardy fields:
Question. What linear orderings can be represented by the set of quasianalytic solutions of (3.3) lying in a common Hardy field?
M. Aschenbrenner, L. van den Dries, and J. van der Hoeven. Asymptotic Differential Algebra and Model Theory of Transseries. Number 195 in Annals of Mathematics studies. Princeton University Press, 2017.
M. Aschenbrenner, L. van den Dries, and J. van der Hoeven. Filling gaps in Hardy fields. Technical Report, Gödel Institute, UIUC, C.N.R.S., 2022.
V. Bagayoko. Hyperexponentially closed fields.
Technical Report, UMons,
V. Bagayoko and J. van der Hoeven. The hyperserial
field of surreal numbers. Technical Report, UMons,
M. Boshernitzan. Hardy fields and existence of transexponential functions. Aequationes mathematicae, 30:258–280, 1986.
B. Dahn and P. Göring. Notes on exponentiallogarithmic terms. Fundamenta Mathematicae, 127(1):45–50, 1987.
J. Écalle. Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac. Actualités Mathématiques. Hermann, 1992.
A. Fornasiero. Recursive definitions on surreal numbers. ArXiv:math/0612234v1, 2018.
H. Gonshor. An Introduction to the Theory of Surreal Numbers. Cambridge Univ. Press, 1986.
H. Kneser. Reelle analytische Lösung der Gleichung und erwandter Funktionalgleichungen. Journal Für Die Reine Und Angewandte Mathematik, 1950:56–67, 01 1950.
D. Marker and C. Miller. Levelled ominimal structures. Revista Matematica de la Universidad Complutense de Madrid, 10, 1997.
A. Padgett. Sublogarithmictransexponential series. PhD thesis, Berkeley, 2022.