|
Conway's class |
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
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
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 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.
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
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(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 log-atomic numbers, i.e. numbers
such that
for all
.
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
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 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.
![]() |
(3) |
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
![]() |
(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 has a cut equation
such that for all
, we have
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 bi-ordered, i.e. that we have
![]() |
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acting on ![]() ![]() |
![]() |
![]() |
![]() |
acting on ![]() ![]() |
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acting on ![]() ![]() |
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 log-atomic 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 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
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 well-defined.
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
.
Moreover, this cut equation is uniform.
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
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) |
is strictly increasing.
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
log-atomic.
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
.
, we have
.
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
.
is bijective. Its
functional inverse
satisfies the following
uniform cut equation on
:
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 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
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 log-atomic
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.
, we
have
.
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
where is infinitesimal. So
, whence
.
Proof. Write and
where
and let
with
. Writing
for
,
we have
. We deduce that
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||
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is strictly increasing
on
.
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
.
is bijective, with
functional inverse
.
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
.
and
with
, the sum
converges to
.
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 well-based and infinitesimal.
Let us now introduce the notion of simplicity for surreal-valued functions.
be a subclass of
and let
be a class of
functions
. We say that a
function
is simplest
if for any
with
and
for any
-minimal
with
, we
have
.
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).
be the class of functions
such that
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).
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. On numbers, germs, and transseries. In Proc. Int. Cong. of Math. 2018, volume 1, pages 1–24. Rio de Janeiro, 2018.
V. Bagayoko and J. van der Hoeven. Surreal substructures. HAL-02151377 (pre-print), 2019.
A. Berarducci and V. Mantova. Surreal numbers, derivations and transseries. JEMS, 20(2):339–390, 2018.
A. Berarducci and V. Mantova. Transseries as germs of surreal functions. Trans. of the AMS, 371:3549–3592, 2019.
M. Boshernitzan. Hardy fields and existence of transexponential functions. Aequationes mathematicae, 30:258–280, 1986.
L. van den Dries and Ph. Ehrlich. Fields of surreal numbers and exponentiation. Fundamenta Mathematicae, 167(2):173–188, 2001.
L. van den Dries, J. van der Hoeven, and E. Kaplan. Logarithmic hyperseries. Trans. of the AMS, 372(7):5199–5241, 2019.
J. Écalle. Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac. Hermann, collection: Actualités mathématiques, 1992.
H. Gonshor. An Introduction to the Theory of Surreal Numbers. Cambridge Univ. Press, 1986.
J. van der Hoeven. Automatic asymptotics. PhD thesis, École polytechnique, Palaiseau, France, 1997.
J. van der Hoeven. Operators on generalized power series. Journal of the Univ. of Illinois, 45(4):1161–1190, 2001.
J. van der Hoeven. Transseries and real differential algebra, volume 1888 of Lecture Notes in Mathematics. Springer-Verlag, 2006.
H. Kneser. Reelle analytische lösung der
gleichung und verwandter
funktionalgleichungen. Journal Für Die Reine Und
Angewandte Mathematik, 1950:56–67, 01 1950.
S. Kuhlmann and M. Matusinski. The exponential-logarithmic equivalence classes of surreal numbers. Order 32, pages 53–68, 2015.
B. H. Neumann. On ordered division rings. Trans. A.M.S., 66:202–252, 1949.
M. C. Schmeling. Corps de transséries. PhD thesis, Université Paris-VII, 2001.