Dendromorphic functions  |
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| Draft version, August 22, 2025 |
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. This work has
been supported by an ERC-2023-ADG grant for the ODELIX project
(number 101142171).
Funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. |
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. This article has
been written using GNU TeXmacs [3].
We study analytic properties of the Picard-Vessiot closure of
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Let 
be a differential field of characteristic
zero whose field of constants is algebraically closed. We say that is Picard-Vessiot closed if any differential
equation
with 
and 
has a
fundamental system of 
linearly independent
solutions over the constant field of .
For any differential field ,
and up to isomorphism, there exists a smallest Picard-Vessiot closed
extension 
that contains
and that has the same constant field as .
We refer to [7] for the algebraic theory of Picard-Vessiot
extensions. In [4], elements of 
are
called differentially definable functions and algorithms are presented
to compute with such functions.
Seidenberg's embedding theorem states that any countably generated
differential field can be embedded into a field of meromorphic functions
on some domain [9, 6]. However, this domain
can be quite small. If 
or 
, then the main goal of this note is to show that
elements of 
can be materialized as analytic
functions on suitable, much larger, Riemann surfaces and to investigate
analytic properties of these functions.
We first explore the kind of Riemann surfaces on which elements of are defined. Intuitively, the only singularities that
can occur are isolated ones, or accumulation points of isolated
singularities, or accumulation points of accumulation points of isolated
singularities, and so on. This leads to the notion of “recursive
discrete ramifications” that will be formally defined in section
2. For simplicity, we will restrict our attention to simply
connected Riemann surfaces, but we note that it should not be hard to
extend the theory to arbitrary connected Riemann surfaces.
Let 
be a simply connected Riemann surface. The
recursively discretely ramified Riemann surfaces above
give rise to an inductive system of analytic function spaces on these
surfaces. The inductive limit 
of these spaces is
the space of dendromorphic functions on . If 
,
then we will show in section 3 that
is a Picard-Vessiot closed field. In particular, 
.
In section 4, we will derive some consequences of the fact
that the only singularities of dendromorphic functions arise as
recursive accumulation points of isolated singularities. In particular,
such singularities cannot give rise to natural boundaries in a strong
sense. As a corollary, we shall see that the generating series of the
number of partitions of an integer does not belong to 
. This answers open question 4 from [1].
The set 
is closed under composition [4].
In section 5, we shall show that this also holds for 
and a suitable class of “boundaryless”
functions. We no not know whether is closed
under functional inversion. Weierstrass 
functions are examples of differentially algebraic dendromorphic
functions that are not in .
Acknowledgments. We are grateful to Gleb Pogudin for his helpful comments on a first version of this note.
Let be a connected Riemann surface. A
Riemann surface above is a pair 
, where 
is
another Riemann surface and 
a holomorphic covering: for every 
,
there exists an open neighborhood 
and a
countable set 
such that 
and 
is a homeomorphism for every 
.
We recall that there exists a Riemann surface 
above with the property that for any other
Riemann surface 
above , there exists a unique 
with
and such that 
is a
Riemann surface above . In
particular, if has the same universal property
as , then 
is a homeomorphism. In other words, the space is
unique up to such a homeomorphism and we call it the covering
space of . We also
recall that the covering space of 
is 
.
Assume now that is a simply connected Riemann
surface and consider a discrete subset 
,
i.e. every 
has an open neighborhood
with 
.
Let 
and consider the covering space 
of 
. We define
and 
,
where 
is the inclusion map, and call 
a discrete ramification of . We will also say that the map 
is a discrete ramification.
and 
of 
. Let 
and 
.
Then 
and, after identification of 
, and 
via these
isomorphisms, the following diagram commutes
Proof. Since 
is a
Riemann surface above 
, there
exists a unique 
with 
. Conversely, 
is a Riemann
surface above 
, so there
exists a unique 
with 
. Next 
is a Riemann surface
above 
, so there exists a
unique map 
with 
.
This shows that and 
are
mutual inverses and 
.
Identifying and 
via this
isomorphism, we have 
and we already showed above
that 
. We conclude that the
top triangle in the above diagram commutes. The isomorphism 
and the commutation of the bottom triangle are proved
similarly.
Now consider a sequence 
with 
and 
for 
,
where 
is a discrete subset of 
. Then we call 
a
recursive discrete ramification of of
height 
. We will also say
that the map 
is a recursive discrete
ramification. If 
are two recursive discrete
ramifications, then we will say that lies
over 
if there exists a recursive discrete
ramification 
with 
.
and of respective heights 
and
, there exists a recursive
discrete ramification of height 
that lies above both and .
Proof. Setting 
,
there exist sequences of discrete ramifications
whose compositions are and , respectively. Using 
applications of Lemma 1, we can extend these sequences into
a commutative diagram
where all the arrows are discrete ramifications. The composite map 
is the desired recursive ramification . Since we can obtain it by following 
diagonal arrows and 
horizontal or
vertical arrows, the height of is 
, as claimed.
Given a simply connected Riemann surface ,
let 
denote the space of analytic functions on
. For any discrete
ramification , we have a
natural injection of into 
into 
. Consequently, any
recursive discrete ramification induces a
natural injection 
. By Lemma
2, we know that these injections form an inductive system.
We denote the inductive limit of these injections by
and call the elements of dendromorphic
functions on .
Any dendromorphic function can concretely be represented as an analytic
function 
for some recursive discrete
ramification . Now consider
two such representations 
and 
for recursive discrete ramifications 
and 
. Let be a
recursive discrete ramification over both 
and
, and let 
and 
be the natural induced injections as above.
Then 
and 
represent the
same dendromorphic function whenever 
.
be a simply connected open subset of 
. Then the space 
is a Picard-Vessiot closed field.
Proof. The space is
clearly a differential ring, since it is the inductive limit of
differential rings 
. Consider
a non-zero dendromorphic function, represented by an analytic function
. Then the set 
of points where 
vanishes is
discrete, so 
is defined on 
, and 
.
Since 
embeds into ,
this shows that is a field. Let us next consider
a differential equation
 |
(1) |
where 
with 
.
By Lemma 2, we may represent 
by
analytic functions in for some recursive
discrete ramification 
. The
set of points where 
vanishes is discrete and it is classical that any solution of (1)
at a point 
can be analytically continued along
any path on that avoids 
. Doing this for a fundamental system of solutions
at 
, we obtain a fundamental
system of solutions 
. We
conclude that is Picard-Vessiot closed.
Let be a simply connected open subset of 
and consider a dendromorphic function 
. We say that 
is
boundaryless if the following property holds for every
representation , where 
is a recursive discrete ramification: given a
continuous path 
with 
for
some 
and 
,
there exists a continuous path 
with 
and
Proof. With ,
, , and 
as in the above
definition of boundaryless, there exist discrete ramifications 
with 
, 
, and 
.
We proceed by induction over .
The result is clear for 
, so
assume that 
and let 
. The induction hypothesis implies the existence of
a continuous path 
with 
and 
. Let
be the discrete subset of 
such that 
.
Since 
is compact, there exists an 
such that for every 
,
the closed ball 
with center 
and radius 
is an isomorphic lift of the ball
. Then the thickened image
of 
is also compact, so
the set 
is finite, since
is discrete. Modulo taking a smaller ,
we may therefore assume that 
and 
for all 
, and
that 
whenever 
.
Let 
be the values of for
which 
. For some sufficiently
small 
with 
,
the intervals 
are pairwise disjoint and 
whenever 
.
Now consider any path 
with the following
properties:
If 
, then 
.
If 
, then 
restricted to 
is any path from
to 
inside the
punctured disk 
.
If 
, then 
for all 
.
By construction, we have 
,
and 
.
Consequently, there exists a unique lift 
with
and 
.
This lift satisfies 
and 
.
Recall that the generating function for the number 
of partitions of an integer 
is given by the
explicit formula
 |
(2) |
It is well known [10, 5] that 
has a natural boundary on the unit circle and that it satisfies an
algebraic differential over .
Consequently, if contains the closed unit disk,
then cannot be boundaryless above .
contains the closed unit disk, then the
function from .
does not belong to 
.
Given a simply connected Riemann surface, we say that a local analytic
function defined at is
dendromorphic on if it lifts and
extends into an analytic function on for some
recursive discrete ramification of .
and 
be two local
analytic functions above 
,
which are both dendromorphic. If 
(after the
natural identification of with its lift), then
is dendromorphic.
Proof. The result is clear if 
, so assume that is
non-constant. There exist sequences of discrete ramifications
of 
and distinguished points 
and 
above that we
identify with , such that the
local functions and
extend analytically to 
and 
, respectively. Let us show how to extend the
first sequence of discrete ramifications into a sequence
where 
again comes with a distinguished lift of
, and such that can be lifted into an analytic map 
that we will also denote by .
Consequently, the composition will be naturally
defined on .
We proceed by induction and note that the inductive property is
trivially satisfied if 
. So
assume that 
and that 
has
been constructed. Let 
be such that 
. By the induction hypothesis, we may regard
as an analytic function from
into 
. Since
is non-constant, the set 
is discrete. We take
and also pick any lift 
of 
to be our distinguished point above zero. By
our choice of , the function
has a unique lift 
with
. This completes our
inductive construction of and thereby also the
proof.
We say that a local analytic function defined at
is boundaryless above an open set 
if for every path with 
and every ,
there exists a path 
with 
and 
, such that can be continued analytically along 
. We simply say that is
boundaryless if it is boundaryless above .
and be two local
analytic functions at 
,
which are both boundaryless. If (after the
natural identification of with its lift), then
is again boundaryless.
Proof. The result is clear if , so assume that 
.
Modulo replacing by 
, where 
is the valuation of
at zero, we may assume without loss of
generality that 
.
Let 
and consider a path 
with . Since
is boundaryless, there exists a path 
with 
and 
, such
that 
is defined on 
.
Since the image of 
is compact, it contains only
a finite number of zeros of 
.
By applying Theorem 4 to 
,
we may also arrange that does not vanish on
. Consequently, the local
functional inverse 
of at
zero can be continued analytically on .
Let 
with be such that
is defined on the compact set
and such that is defined on 
. Let
Since is boundaryless, there exists a path 
with 
and 
, such that 
is defined
on . But then 
, 
,
and 
is defined on .
We finally observe that 
.
It would be interesting to know whether is also
closed under functional inversion. The functional inverse of any entire
function is easily seen to be in .
Any Weierstrass elliptic function 
is also in
, since it is meromorphic on
. The function is not holonomic, but it is the functional inverse of a
holonomic function. In fact, is differentially
algebraic, but not in . This
is due to the fact that embeds into any
differentially valued field and Picard-Vessiot closed field 
of complex transseries from [2], but such a
field never contains
(see also [8, Example 9]).
The functional inverse of from (2)
might be a candidate for a dendromorphic function
whose functional inverse is not. We expect this functional inverse to
have a very dendromorphic-like structure, although it might necessitate
a non-finite number of discrete ramifications in the required global
sense. Variations on the concept of dendromorphic functions might
therefore be another topic for further investigations.
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