Conway's class |
|
Conway's class |
Let 
be Conway's class of surreal numbers. It is
well known that admits 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. Following Conway's
tradition, all numbers will understood to be surreal in what
follows.
The aim of this note is to define a bijective function 
on the class 
of positive infinitely large
numbers, which is strictly increasing and satisfies the functional
equation
 |
(1) |
for all 
. Since this equation
admits many solutions, the main difficulty is to single out a particular
solution that will be most “natural” in a way that needs to
be made precise.
The function 
is said hyperexponential
and it is the first non-trivial hyperexponential in the transfinite
sequence 
of iterated exponentials
The corresponding reciprocals are called hyperlogarithms:
It is natural to require such more general hyperexponentials to satisfy
for all ordinals 
and
. Similarly, 
.
There are known real analytic solutions of (1) with good
properties [12, 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 [14]:
they generalized transseries to include formal counterparts 
, 
of 
, 
for 
[14]. 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 [7] constructed the
field of logarithmic hyperseries 
with
for all ,
with corresponding natural hyperlogarithms 
.
The ultimate goal [11, 2] is to produce a
field of hyperseries 
with all hyperexponentials
and to construct an isomorphism 
.
This work can be considered as another step in this direction, by
constructing a natural 
.
can be
written as infinite series
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 [14];
see [4].
We will freely use notations from [1, 11] when
dealing with such transseries. In particular, the support of
is defined as 
and its
infinite part as 
.
Here we used Hardy's notation 
for 
. Similarly, we set 
. We also define 
,
, and 
. Finally, we write 
if 
, in which case we say that 
is a truncation of 
.
and decompose it as 
with 
,
, and 
. Then the functional equation of yields
 |
(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 
.
,
, , and are examples of
so-called surreal substructures, which were extensively studied in [3]. Let us quickly recall a few basic facts; see also
section 2.
Besides the usual ordering, the class of surreal
numbers admits 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 
. 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 as 
), the
set 
(resp. 
) is cofinal (resp. coinitial) with
respect to 
(resp. 
).
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.
has been defined for some positive
purely infinite number 
.
For sufficiently small 
, we
wish to define 
using Taylor series expansion:
 |
(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
. More generally, consider
with 
for a certain 
. Assuming (1), this
means that 
, which allows us
to define
 |
(4) |
and set
 |
(5) |
It follows that it is sufficient to define at
numbers 
with
where 
. 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.
proceeds in three stages:
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)
when extending 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 to define surreal substructures using
convex partitions. For more details, see [3, Section 6].
be a surreal substructure and 
be a function. Let 
be functions
defined for cut representations in and such
that 
is a cut representation in whenever is a cut representation
in . We say that 
is an equation of 
if,
for all , we have
We say that the equation is uniform if we have 
whenever is a cut
representation in . For
instance, by [9, Theorem 3.2], for 
, the following equation for the translation 
by 
is uniform:
 |
(6) |
Remark 
has an equation 
with
This is in particular the case 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 
.
be a surreal substructure and let 
be a partition of into
convex subclasses, each of which admits a cofinal and coinitial
subset. We refer to as a thin convex
partition of . For
, we let 
denote the unique member of containing . We also write 
for any subclass 
of . We say that an element
is -simple if it
is the -minimum of its
class . This is equivalent
to the existence of a cut in
with and 
.
Then the class 
of -simple
elements forms a surreal substructure which is contained in . For 
,
we have 
.
on a surreal
substructure is a group of strictly increasing
bijections 
under functional composition. We
regard elements 
of as
actions on and sometimes write 
and 
for
rather than 
and 
.
For such a function group ,
the collection 
of classes
is a thin convex partition of 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
for actions of the following function groups
acting on ,
or .
For and 
,
we define
 |
 |
 |
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 [13].
For , we will denote 
the unique log-atomic element of 
and we will denote 
the unique element of 
lying in .
We have 
.
Let 
denote the field of logarithmic hyperseries
from [7]. For each ,
we define 
. Recall 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 .
Proof. Assume for contradiction that 
. Write 
and
let 
be minimal with 
. Since 
is Noetherian as a
series in , the set 
is well based and admits a largest element 
. Taking 
sufficiently
small such that 
, it follows
that 
, whence 
.
Proof. The left and right hand sides of (7) are clearly Noetherian series in . For any 
in 
, the following Taylor series expansions hold
in 
:
Subtracting both expansions, the identity (7) holds for
substituted by 
in . We conclude by Lemma 2.
In [7], we defined the field 
,
as well as a hyperlogarithmic function on 
for which 
with 
. 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
whence
For 
in ,
this allows us to define 
using the Taylor series
expansion
 |
(8) |
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 inverse
functions. For a detailed proof, we refer to [14, section
6.4]. We conclude by Lemma 2.
We recursively define for positive purely
infinite numbers by
defines a
strictly increasing bijection 
.
Moreover, the previous 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
for
form a thin convex partition of .
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.
is a surreal substructure.
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 
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 use
recursion on to define
 |
(11) |
is a strictly increasing function
.
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.
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 .
, 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 
.
is bijective. Its
reciprocal admits the following uniform
equation on 
:
Proof. Noticing that 
for
all , this follows from the
same arguments as in Proposition 5.
The field of logarithmic hyperseries of [7] 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 [8, 10,
14]. 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
and
Proof. If 
,
then this is Proposition 9. Otherwise, we have 
and 
, whence
and
Inversely, consider an arbitrary positive infinite number 
. Then there exists a
such that 
for some log-atomic 
and 
. 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 .
Proof. With as above
(while taking 
), we have
where 
because of Proposition 9.
, 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 4, it therefore follows
that 
inside 
.
The result follows by specializing this relation at . If ,
then 
by Proposition 13. Applying
the result for the special case when ,
we have 
. We conclude by
Proposition 12.
In particular, the function is surjective. We
next prove that it is strictly increasing, concluding our proof that
is a strictly increasing bijection with
reciprocal .
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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is strictly increasing
on .
Proof. Let with 
. If 
,
then we get 
by Lemma 15. Otherwise,
we have 
so by Lemma 16, there are
and with
Since 
, we conclude that
, whence 
.
is bijective, with
reciprocal .
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with good properties.
such that the substitution 
of 
.
and 
,
.
with 
,
.
,