Defining a surreal hyperexponential

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

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

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 surreal-analyticity [5], i.e. the existence of Taylor expansions around every point. In our final section, we prove that our function

is the simplest surreal-analytic 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 non-trivial hyperexponential in the transfinite sequence

of iterated exponentials

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 surreal-valued 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.

Surreal numbers and transseries

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

Defining exponentials

where

is the usual exponential in

and

. In order to define

, 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 log-atomic numbers, i.e. numbers

such that

for all

Surreal substructures

Besides the usual ordering, the class

of surreal numbers comes with a well-founded partial order

called the simplicity relation. A surreal substructure is a subclass

that is isomorphic to

for the induced relations by

and

. Equivalently, this means that for any subsets

with

, the cut

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

containing

such that

, the set

(resp.

) is cofinal (resp. coinitial) with respect to

(resp.

Surreal substructures are particularly well-suited for defining functions via well-founded 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.

Truncated numbers

The successive derivatives

can be defined in

as ordinary (and so-called 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

It will thus be sufficient (see the discussion after Corollary 8) to define

at all numbers

with

where

, and then extend

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.

Defining hyperexponentials

2.Equations and convex partitions

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.

Uniform cut equations

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

is uniform:

This is trivially the case for

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

with

, and we have

. 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

and

yield strictly increasing functions.

Convex partitions

Then the class

-simple elements forms a surreal substructure which is contained in

[3, Theorem 6.7]. For

, we have

[3, Proposition 6.8].

Function groups

For

, the relation

is a partial order on

. We will frequently rely on the elementary fact that

is partially bi-ordered, i.e. that we have

Common function groups

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 log-atomic element of

and

to be the unique element of

inside

. We have

, by definition.

3.Logarithmic transseries and hyperseries

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

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 well-based). In particular, the support of any non-zero 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

(i.e.

), one may then define the substitution

[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

, the following Taylor series expansions hold [9, Proposition 8.1] in

Subtracting both expansions, the identity (7) holds for

substituted by

. We conclude by Lemma 2.

In [9, Section 7], it was shown that there is a hyperlogarithmic function

for which

. Let

. Then logarithmic transseries in

can be considered as elements in

and the successive derivatives of

with respect to

are given by

We now give two lemmas that will be used to extend

into a bijective function via Taylor series expansions.

where

is infinitesimal. It follows from Neumann's theorems [17, Theorems 3.4 and 3.5] that the sum

is well-defined.

Proof. The left hand side is well defined by (8) for

. The fact that (9) holds for

substituted by

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.

4.Hyperexponentials of truncated numbers

In this section, we deal with step 1 and 2 of the definition of

. As outlined in the introduction, we first define

(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).

Proof. The function

is well-defined 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

(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

Proof. Since

is a surreal substructure, the definition, strict monotonicity and uniformity follow by Remark 1. For

, we have

since

. We deduce that

-simple, hence log-atomic.

We now show that

satisfies the functional equation (1). By [4, Lemma 2.4], for every infinite monomial

, we have

Proof. We prove this by induction on

. Let

such that this holds on

. Note that

for all

and

. We have

, so

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

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

. 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.

5.Hyperexponentials of arbitrary numbers

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 well-based 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

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 log-atomic

and

[4, Corollary 5.11]. We extend the definition of

to any such number

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

, 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. Let

be such that

, where

and

. Let us first consider the special case when

. Since

is log-atomic, 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

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

6.Simplicity of

We conclude this paper by showing that

is the simplest well-behaved solution to (1) in a sense that will be made precise. Let us first show that

is surreal-analytic. 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

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

In general, let

be minimal with

. We will prove the result by induction on

. In view of the special case, we have

As in the proof of [5, Proposition 3.16], it is enough to justify the convergence of the sum

The above definition was first introduced in [4] in order to show the existence of a “simplest pre-derivation” 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).

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

. 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

) so

lies in the cut

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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1.Introduction

Surreal numbers and transseries

Defining exponentials

Surreal substructures

Truncated numbers

Defining hyperexponentials

2.Equations and convex partitions

Uniform cut equations

Convex partitions

Function groups

Common function groups

3.Logarithmic transseries and hyperseries

4.Hyperexponentials of truncated numbers

5.Hyperexponentials of arbitrary numbers

6.Simplicity of

Bibliography