
Conway's class of surreal numbers has a rich structure: it forms a real closed field with an exponential function and a derivation. The aim of this note is to construct a surrealvalued solution to the functional equation with good properties. 
Let be Conway's class of surreal numbers [7]. It is well known that has a rich structure: Conway showed that forms a real closed field and Gonshor also defined an exponential function on that satisfies the same first order theory as the usual exponential function on the reals [11, 8]. Following Conway's tradition, all numbers will be understood to be surreal in what follows.
Let be the class of positive infinitely large numbers, i.e. numbers that are larger than any integer. The aim of this note is to define an increasing and bijective function that satisfies the functional equation
(1) 
for all . Since this equation has many solutions, one difficulty is to single out a particular solution. Now one interesting property for surreal functions is surrealanalyticity [5], i.e. the existence of Taylor expansions around every point. In our final section, we prove that our function is the simplest surrealanalytic solution to (1) in a sense that will be made precise in Definition 21.
The function is called a hyperexponential and it is the first nontrivial hyperexponential in the transfinite sequence of iterated exponentials
The corresponding functional inverses are called hyperlogarithms:
We require such more general hyperexponentials to satisfy for all ordinals and . Similarly, .
There are known real analytic solutions of (1) with good properties [15, 6], even though there does not seem to exist any meaningful “most natural” solution. It is also well known that fractional iterates of and can be defined in terms of and : given , we take and .
From a formal perspective, hyperexponentials and hyperseries were studied in detail by Schmeling and van der Hoeven [18]: they generalize transseries to include formal counterparts of , for [18]. This yields in particular a natural hyperexponential on the set of positive infinitely large transseries. More recently, van den Dries, van der Hoeven and Kaplan [9] constructed the field of logarithmic hyperseries with hyperlogarithms for all . The ultimate goal [14, 2] is to construct a field of hyperseries that is also closed under all hyperexponentials, together with an isomorphism that corresponds to evaluation of hyperseries at . This work can be considered as another step in this direction, by constructing the first surrealvalued hyperexponential function . The higher hyperexponentials with can be constructed in a similar way, but this more technical generalization will be the subject of a forthcoming paper.
The remainder of this introduction starts with some quick reminders about surreal numbers, transseries, and surreal substructures. We next outline the main ideas of our construction, while highlighting a few similarities with the construction of the usual exponential function.
where denotes the class of surreal monomials and the coefficients are real. In particular, is isomorphic to the Hahn field of formal power series. Together with the exponential function, even admits the structure of a field of transseries in the sense of [18]; see [4]. Our definition of will rely on this existing structure on , and in particular on the fact that certain transseries act as functions on surreal numbers [5].
We will freely use notations from [1, 14] when dealing with such transseries. In particular given , we define
The set is called the support of ; it is always a set (as opposed to a proper class) and it is reverse well ordered. For we set
When , we say that is a truncation of . We also define
(2) 
where is the usual exponential in and . In order to define on , this relation shows that it would have sufficed to define it on . In addition, it can be shown that bijectively maps the class to the class . Our process to define is similar, with different subclasses and in the roles of and . The class is defined below and is the class of logatomic numbers, i.e. numbers such that for all .
Besides the usual ordering, the class of surreal numbers comes with a wellfounded partial order called the simplicity relation. A surreal substructure is a subclass of that is isomorphic to for the induced relations by and on . Equivalently, this means that for any subsets of with , the cut
in admits a minimum which is then denoted by . We call a cut representation in . We extend this notation to the case when are classes, provided indeed exists. If , then we let
Then . Moreover, for any cut in containing such that , the set (resp. ) is cofinal (resp. coinitial) with respect to (resp. ).
Surreal substructures are particularly wellsuited for defining functions via wellfounded induction on , as will be the case for certain restrictions of . Important examples of surreal substructures include , the class of strictly positive numbers, the class of positive infinitely large numbers, the classes and of monomials and infinite monomials, the class of purely infinite numbers, and the class of infinitesimal numbers.
(3) 
The successive derivatives can be defined in as ordinary (and socalled logarithmic) transseries applied to :
It can be shown that (3) converges formally, provided that (see Lemma 4 below). More generally, consider with for a certain . Assuming (1), this means that . We will see that this inequality allows us to define
(4) 
In view of (1), one must then set
(5) 
It will thus be sufficient (see the discussion after Corollary 8) to define at all numbers with
where , and then extend at for as in (4) and (5). Those numbers are said to be truncated and we will write for the class of all truncated numbers. It turns out that is a surreal substructure.
We first define on . For any two positive purely infinite numbers with , we have . By the functional equation, we should have . We deduce that for , the number should lie in the cut in where
The simplest way to ensure this is to define
for all .
We next extend to . Similar arguments and the simplicity heuristic impose
where is a function group to be defined in Section 2. Here , , and (3) play a similar role as , , and (2) for the definition of .
We finally extend the definition of to by relying on (4) and (5).
Before we define , let us briefly recall a general method from [3, Section 6] to define surreal substructures using convex partitions and function groups. In section 4, we will use this to show that and are indeed surreal substructures.
We say that the cut equation is uniform if we have whenever is a cut representation in . For instance, by [11, Theorem 3.2], for , the following cut equation for the translation by is uniform:
(6) 
Remark
This is trivially the case for if and for all . Then we claim that is strictly increasing. To see this, consider with . By [3, Proposition 4.6], there is a maximal element of with , and we have or . We treat the first case, the other one being symmetric. Since and , we have so there is with . We have . A similar argument yields , so .
Note that the existence of relies on the fact that is a surreal substructure. We will use this result in order to deduce that our definitions of on and yield strictly increasing functions.
Then the class of simple elements forms a surreal substructure which is contained in [3, Theorem 6.7]. For , we have [3, Proposition 6.8].
is a thin convex partition of [3, Proposition 6.25] and we define .
For , the relation is a partial order on . We will frequently rely on the elementary fact that is partially biordered, i.e. that we have
acting on or  
acting on or  
acting on or 
We then have the following list of identities [3, Section 7.1]:
The action of on (resp. ) yields (resp. ).
The action of on (resp. ) yields (resp. ).
The action of on yields .
The action of on yields [4, Corollary 5.17].
The action of on yields the class of [16].
The groups , will play an important role in this paper. Here we note the analogy between the roles of with respect to and with respect to . The above function groups and identities also turn out to be convenient for asymptotic growth computations that involve , , , and . For , we define to be the unique logatomic element of and to be the unique element of inside . We have , by definition.
Let us briefly recall some basic definitions and facts about logarithmic hyperseries from [9]. Given an ordinal , let be the set of formal power products , for some function . This set carries a natural group structure and we define the monomial ordering on by setting if and only if and for . We call the monomial group of logarithmic hypermonomials of force .
Since is a monomial group, we may form the Hahn field and call it the field of logarithmic hyperseries of force . The elements of are formal sums for which the support contains no infinite strictly increasing sequences for (we say that is wellbased). In particular, the support of any nonzero series has a maximal element that we call the dominant monomial of . We also declare to be positive if and this gives the structure of a real closed field.
The ordered field is also equipped with a derivation whose values at monomials are given by
where for all . This derivation is strongly linear in the sense that it preserves infinite sums, whenever defined.
In [13], the theory of Hahn fields was generalized to partially ordered monomial monoids and series with Noetherian support (no infinite increasing sequences and no infinite antichains). Recall from there that stands for the set of series with Noetherian support in the partially ordered set . We may consider elements of as bivariate series that are logarithmic transseries with respect to and ordinary series with respect to . Given and in (i.e. or ), one may then define the substitution of by in [13, section 3.4]. The following lemma will be very useful for proving formal identities in .
Proof. Assume for contradiction that . Write and let be minimal with . Since is Noetherian as a series in , the set admits a largest element . Taking sufficiently small such that , it follows that , whence .
The following lemma will be used later in order to prove that the functional inverse of the function is defined.
Proof. The left and right hand sides of (7) are clearly Noetherian series in . For any in , the following Taylor series expansions hold [9, Proposition 8.1] in :
Subtracting both expansions, the identity (7) holds for substituted by in . We conclude by Lemma 2.
In [9, Section 7], it was shown that there is a hyperlogarithmic function on for which . Let . Then logarithmic transseries in can be considered as elements in and the successive derivatives of with respect to are given by
For any , we have
where
whence
We now give two lemmas that will be used to extend into a bijective function via Taylor series expansions.
Proof. For , we have
where is infinitesimal. It follows from Neumann's theorems [17, Theorems 3.4 and 3.5] that the sum is welldefined.
Proof. The left hand side is well defined by (8) for in . The fact that (9) holds for substituted by in follows from the usual rules of iterated derivatives of functional inverses. For a detailed proof, we refer to [18, section 6.4]. We conclude by Lemma 2.
In this section, we deal with step 1 and 2 of the definition of . As outlined in the introduction, we first define on (Proposition 6) and then on (Proposition 9). This second definition will require identifying and showing that it is a surreal substructure (Corollary 8) by means of convex partitions (Proposition 7).
We first recursively define for positive purely infinite numbers by
Proof. The function is welldefined and strictly increasing by Remark 1. The uniformity of the equation follows immediately.
Let denote the partial inverse function of and prove that is defined on by induction on . Let such that is contained . Since is injective, its inverse is defined on . Let
This number is well defined since is strictly increasing and for , we have . By uniformity, we have where . In order to conclude that , it therefore suffices to show that lies in the cut . We have by (10) and by definition of , whence since . We conclude by induction that is surjective.
We next identify the class of truncated numbers. For , we consider the following convex class
Proof. Given , it is clear that the class is convex and that it contains . Note that for , we have . Let with . We claim that . If , then we have , which yields the result. Assume that . Assume for contradiction that there are and with and . Given , there is a number with . Therefore are dominated by , whence . This proves that and symmetric arguments yield : a contradiction. This proves our claim. It only remains to see that the class admits a cofinal and coinitial subset for any . Indeed, we can take as examples of such sets.
Let and let denote the supremum of truncations of (i.e. series with ) with . In particular, we have since . We see that satisfies and . Write and . By maximality of , we have for some , so , or equivalently . We deduce that is the minimum, hence minimum of , so . We also see that for and , we have . Since is a surreal substructure, we may recursively define for by
(11) 
Proof. Since is a surreal substructure, the definition, strict monotonicity and uniformity follow by Remark 1. For , we have since on . We deduce that is simple, hence logatomic.
We now show that satisfies the functional equation (1). By [4, Lemma 2.4], for every infinite monomial , we have
(12) 
Proof. We prove this by induction on . Let such that this holds on . Note that for all and . We have , so
We deduce that
Since , we may apply (12). We also note that and are mutually cofinal and coinitial for all to obtain
We have , so and . We clearly have . We deduce that . By induction, the relation is valid on .
It remains to check that the function extends .
Proof. We prove this by induction on . Let be such that this holds on . For , we have , and there is with . We deduce that . In particular, we have so . This proves that is cofinal with respect to . For and , we have where , so is cofinal with respect to . Symmetric arguments yield that and are mutually coinitial. We conclude that .
Since and agree on , we will identify both functions hereafter and simply write instead of . Let us finally examine its functional inverse .
Proof. Noting that for all , it suffices to show that is strictly increasing and its cut equation is uniform, which follows from the same arguments as in Proposition 6.
In this section, we complete the final step 3 of the definition of , by extending its domain to the whole class . The field of logarithmic hyperseries of [9] is a subfield of the class of all wellbased transseries in an infinitely large variable . Both and the class of all transseries are closed under derivation and under composition [10, 12, 18]. For every positive infinite number , there also exists an evaluation embedding such that for all : see [5].
Given , let be the unique truncated series with . If , then there is a smallest number with
Write . With as in section 3, we define for every :
Substitution of for in (8) allows us to extend the definition of by
(13) 
and
Let us first check (1) for this extended definition.
Proof. If , then this is Proposition 10. Otherwise, we have and , whence and
Inversely, consider an arbitrary positive infinite number . Then there exists a such that for some logatomic and [4, Corollary 5.11]. We extend the definition of to any such number by
In view of Lemma 3, the value of does not depend on the choice of . Note also that this definition indeed extends our previous definition of on , since when .
In order to prove that the extended functions and are functional inverses of each other, we first check that satisfies the “inverse” of the functional equation (1).
Proof. With as above (while taking ), we have
where because of Proposition 10.
Proof. Let be such that , where and . Let us first consider the special case when . Since is logatomic, we have . From Lemma 5, it therefore follows that inside . The result follows by specializing this relation at . If , then by Proposition 14. Applying the result for the special case when , we have . We conclude by Proposition 13.
In particular, the function is surjective. We next prove that it is strictly increasing, concluding our proof that is a strictly increasing bijection with functional inverse . For this, we need two lemmas.
Proof. Note that , so it is enough to prove that for all . For such , there is with , and
where is infinitesimal. So , whence .
Proof. Write and where and let with . Writing for , we have . We deduce that
Proof. Let with . If , then we get by Lemma 16. Otherwise, we have so by Lemma 17, there are and with
Since , we conclude that , whence .
We conclude this paper by showing that is the simplest wellbehaved solution to (1) in a sense that will be made precise. Let us first show that is surrealanalytic. Generalizing (13), we let for all and .
Proof. Let . Assume first that . We rely on the notion of horizontal saturation of Taylor families from [18, section 6.2.2]. The restrictions of the functions to form a Taylor family with index . Indeed, for with , we have , so the family is not summable. By [18, Proposition 6.2.4], the family has a minimal horizontally saturated expansion . The domain of in particular contains the class of numbers with and . Given such a number , we have
Indeed, the first equality holds by definition (since ), and the second one by horizontal saturation. In particular, this yields and . For , it follows that
so the sum converges as in (13). By horizontal saturation, we get
whence
This concludes the proof in the special case when .
In general, let be minimal with . We will prove the result by induction on . In view of the special case, we have
Now assume that and let . Then
converges to by the induction hypothesis.
converges to , by using the Taylor expansion of at and the fact that
converges, since .
For and , let
As in the proof of [5, Proposition 3.16], it is enough to justify the convergence of the sum
But this directly follows from the fact that
where the set
is wellbased and infinitesimal.
Let us now introduce the notion of simplicity for surrealvalued functions.
The above definition was first introduced in [4] in order to show the existence of a “simplest prederivation” on surreal numbers. By definition, if there is a simplest function in , then it is unique. We can now formulate our main result about the simplicity of our solution to (1).
is strictly increasing.
for all .
for all and .
For and with , the sum converges to .
Then is the simplest element of .
Proof. Let be such that and let be simplest with . Given and with , we have , by () and (). Taking and with , this implies that , i.e. .
Since is strictly increasing and surjective, the classes with form a convex partition of . Since is injective with , we have . We claim that . Assume for contradiction that there is with . By convexity of , we have or . Set in the first case and in the second case. In both cases, we get . Since is convex, we also have . But then , since ; a contradiction. Applying similar arguments to instead of , one obtains , whence .
Let . We have because and are simple. The previous argument and ) yield . So , and likewise . We have by ) so lies in the cut
Recall that holds by definition, so .
Acknowledgments. The first author is supported by the French Belgian Community through a F.R.I.A. grant. The third author is supported by EPSRC (grant reference EP/T018461/1).
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