# Generalized single-valued hyperlogarithms

Oliver Schnetz  Department Mathematik
Cauerstraße 11
91058 Erlangen, Germany<EMAIL_ADDRESS>
###### Abstract.

Single-valued hyperlogarithms are generalized to include primitives of
differential forms $\mathrm{d}z/(az{\overline{z}}+bz+c{\overline{z}}+d)$,
$a,b,c,d\in{\mathbb{C}}$, where ${\overline{z}}$ is the complex conjugate of
the variable $z\in{\mathbb{C}}$. The construction of these generalized single-
valued hyperlogarithms (GSVHs) relies on a commutative hexagon which allows
one to express primitives as anti-primitives. The article provides a proved
constructive theory of GSVHs.

##  1\. Introduction

Single-valued hyperlogarithms on the punctured complex plane have a plethora
of applications in mathematics. A classical example is the Bloch-Wigner
dilogarithm (see [24] and the references therein),

(1) $D(z)=\mathrm{Im}\,(\mathrm{Li}\,_{2}(z)+\log(1-z)\log|z|),$

where $\mathrm{Li}\,_{2}(z)=\sum_{k=1}^{\infty}z^{k}/k^{2}$ is the
(multivalued) dilogarithm. There exists a natural class of single-valued
functions which generalizes the Block-Wigner dilogarithm to hyperlogarithms of
higher weights. This class of single-valued hyperlogarithms is well studied
and understood (see, e.g., [4, 5, 17]).

In recent years, certain calculations in perturbative Quantum Field Theory
(pQFT) led to single-valued functions on the punctured complex plane (see,
e.g., [13, 17]). It quickly became clear that the class of single-valued
hyperlogarithms is too restricted to describe many of these functions (even if
they are of hyperlogarithmic nature) [11, 17, 18].

While single-valued hyperlogarithms are iterated integrals of differential
forms with poles in ${\mathbb{C}}$,

$\frac{\mathrm{d}z}{z-a},\quad a\in{\mathbb{C}},$

there is demand for including differential forms with denominators which are
bilinear in $z$ and ${\overline{z}}$ (the complex conjugate of $z$),

(2) $\frac{\mathrm{d}z}{az{\overline{z}}+bz+c{\overline{z}}+d},\quad
a,b,c,d\in{\mathbb{C}}.$

One expects that (iterated) single-valued primitives of these differential
forms exist if the zero locus of the denominator is empty in ${\mathbb{C}}$.
By integrating $\mathrm{d}z/(z+1/{\overline{z}})$, e.g., we get
$\log(z{\overline{z}}+1)$ which is single-valued on ${\mathbb{C}}$ (Example
68).

This setup, however, still is too specific. In practice, one often has
denominators which vanish on a curve in ${\mathbb{C}}$. A frequent case is the
denominator $z-{\overline{z}}$ which vanishes on the real axis. In general,
these differential forms have no single-valued primitives on the punctured
complex plane. The logarithm $\log(z-{\overline{z}})$ inherits the singular
real axis from the differential form.

Nonetheless, it may happen that a numerator cancels the singularity of the
denominator to render the function single-valued on the punctured complex
plane. Because $D(z)$ in (1) vanishes on the real line, one examples of this
type is (see Example 71, Section 8.3, and [11])

(3) $\frac{D(z)}{z-{\overline{z}}}.$

Another example of a function with lifted singularity is (Example 69)

(4) $\frac{\log(z{\overline{z}})}{z-1/{\overline{z}}}.$

Both functions have single-valued primitives on the (punctured) complex plane.
Generalized single-valued hyperlogarithms (GSVHs) are the most general class
of single-valued functions which emerge from iteratively integrating
differential forms (2), see Definition 44. In this article we present a theory
of these functions. We prove the existence of single-valued primitives (and
other fundamental properties of GSVHs) in Theorem 74 using the commutativity
of the hexagon in Figure 1. The proof is constructive (Section 7) and
implemented in the Maple package HyperlogProcedures [19] (which also contains
application to pQFT).

In detail, the paper is organized as follows. In Sections 2 and 3 we set up
the notation and compile some results on iterated integrals and
hyperlogarithms. In Section 4 we generalize to hyperlogarithms in two
variables $z$ and ${\overline{z}}$ which can either be considered independent
or complex conjugates. We fix a quadratically closed number field
${\mathbb{F}}\subseteq{\mathbb{C}}$ which is closed under complex conjugation,
$\overline{{\mathbb{F}}}={\mathbb{F}}$, and construct the
${\mathbb{C}}$-vector-space $\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}$ of
(possibly multivalued) generalized hyperlogarithms with a finite singular set
$z\in\Sigma({\overline{z}})\subset\\{-(c{\overline{z}}+d)/(a{\overline{z}}+b),\;a,b,c,d\in{\mathbb{F}}\\}$
which emerges from iteratively integrating differential forms of type (2). In
Section 5 we define the ${\mathbb{C}}$-algebra
$\mathcal{S}\mathcal{V}_{\Sigma_{0}}$ of single-valued functions with point-
like singularities in a finite set $\Sigma_{0}\subset{\mathbb{C}}$ (which – at
this point – is not related to $\Sigma({\overline{z}})$). Our notion of
single-valuedness does not only demand the absence of monodromies but the
(stronger) existence of single-valued log-Laurent expansions

$\sum_{\ell=0}^{L}\sum_{m,{\overline{m}}\geq
M_{a}}c_{\ell,m,{\overline{m}}}^{a}[\log(z-a)({\overline{z}}-{\overline{a}})]^{\ell}(z-a)^{m}({\overline{z}}-{\overline{a}})^{\overline{m}},\quad
M_{a}\in{\mathbb{Z}},$

at all $a\in\Sigma$ and at infinity (Definition 35; these expansions are
natural in the context of pQFT [1, 17, 18]).

Generalized single-valued hyperlogarithms are defined in Section 6 as the
intersection of both spaces

$\mathcal{G}_{\Sigma}^{\mathbb{C}}=\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}\cap\mathcal{S}\mathcal{V}_{\mathbb{C}},\quad\text{where}\quad\mathcal{S}\mathcal{V}_{\mathbb{C}}=\bigcup_{\Sigma_{0}\subset{\mathbb{C}}}\mathcal{S}\mathcal{V}_{\Sigma_{0}}$

is the union of $\mathcal{S}\mathcal{V}_{\Sigma_{0}}$ over all finite singular
loci $\Sigma_{0}$.

Likewise, $\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ is the union over all
$\mathcal{G}_{\Sigma}^{\mathbb{C}}$. We define
$\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$,
$\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$,
$\partial_{z}\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$
which – by Theorem 74 – become derivatives of
$\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$. The functions (3) and (4), e.g., are
in $\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$. These function spaces
are subsets of the general space of GSVHs $\mathcal{G}_{\mathbb{F}}$ where
hyperlogarithms in $\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ may have
coefficients with bilinear denominators in $z$ and ${\overline{z}}$ (which are
consistent with single-valuedness).

With these notations we prove the commutativity of the hexagon in Figure 1,
where projections $\pi_{\partial_{z}}$ and $\pi_{\partial_{\overline{z}}}$
kill (anti-)residues in $\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$
and $\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$,
respectively.

1.0 Black
[arrow,arrowpos=1,arrowlength=5,arrowwidth=2,arrowinset=0.2](48,38)(48,56)
[arrow,arrowpos=1,arrowlength=5,arrowwidth=2,arrowinset=0.2](128,38)(128,56)
[arrow,arrowpos=1,arrowlength=5,arrowwidth=2,arrowinset=0.2](54,17)(71,7)
[arrow,arrowpos=1,arrowlength=5,arrowwidth=2,arrowinset=0.2](122,77)(105,87)
[arrow,arrowpos=1,arrowlength=5,arrowwidth=2,arrowinset=0.2](105,7)(122,17)
[arrow,arrowpos=1,arrowlength=5,arrowwidth=2,arrowinset=0.2](54,77)(71,87)
(42,24)[lb]$\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$(122,24)[lb]$\partial_{{\overline{z}}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$(122,65)[lb]$\partial_{{\overline{z}}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$(42,65)[lb]$\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$(32,43)[lb]$\pi_{\partial_{z}}$(133,43)[lb]$\pi_{\partial_{\overline{z}}}$(40,84)[lb]$\int_{\mathrm{sv}}\mathrm{d}z$(118,84)[lb]$\int_{\mathrm{sv}}\mathrm{d}{\overline{z}}$(77,-5)[lb]$\partial_{{\overline{z}}}\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$(86,88)[lb]$\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$(52,2)[lb]$\partial_{{\overline{z}}}$(117,0)[lb]$\int_{\mathrm{sv}}\mathrm{d}z$

Figure 1. The inductive construction of GSVHs by a commutative hexagon.

We use the commutative hexagon to prove the structure theorem for GSVHs
(Theorems 63 and 74).

######  Theorem 1.

The hexagon in Figure 1 commutes. The space $\mathcal{G}_{\mathbb{F}}$ of
GSVHs is stable under taking (anti-)primitives, complex conjugation, and
fractional linear transformations. For any finite set of fractional linear
transformations $\Sigma({\overline{z}})$ and any finite set $\Sigma_{0}$ of
singular points in ${\mathbb{F}}$ we obtain an exact sequence

(5)
$0\longrightarrow{\mathbb{C}}\longrightarrow\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}\cap\mathcal{S}\mathcal{V}_{\Sigma_{0}}\stackrel{{\scriptstyle\partial_{z}}}{{\longrightarrow}}\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}\cap\mathcal{S}\mathcal{V}_{\Sigma_{0}}\longrightarrow
0.$

Moreover,
$\mathcal{G}_{\Sigma}^{\mathbb{C}}=\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}\cap\mathcal{S}\mathcal{V}_{\Sigma({\overline{z}})\cap{\mathbb{C}}}$.

In the last two sections we investigate GSVHs more concretely. Section 7
explains how to explcitly use the commutative hexagon for the construction of
GSVHs. The results of this section are the basis for the Maple implementation
of GSVHs in HyperlogProcedures [19]. In Section 8 we focus on GSVHs whose
letters in $\Sigma({\overline{z}})$ relate to involutions on the complex
plane. We conclude the article with the special case
$\Sigma({\overline{z}})=\\{0,1,{\overline{z}}\\}$, where the involution is
complex conjugation (Section 8.3). The ${\mathbb{C}}$-vector-space
$\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathbb{C}}$ has finite dimensions at
fixed weights (the number of iterated integrations). We conjecture these
dimensions up to weight nine (Table 1) with a proof up to weight five (Theorem
90). In Theorem 93 we prove that GSVHs in $\\{0,1,{\overline{z}}\\}$ are
defined over the ${\mathbb{Q}}$-algebra of multiple zeta values (MZVs, see
(88)).

## Acknowlegements

The author is indebted to Dirk Kreimer for his support while being a visiting
scientist at Humboldt University, Berlin, from 2011 to 2015 where the project
started and many results were obtained. He is also very grateful to F. Brown
for his interest which lead to many inspiring and helpful discussions. The
author is supported by DFG grant SCHN 1240.

##  2\. Iterated integrals

Let

(6) $\Sigma=\\{s_{1},s_{2},\ldots,s_{N}\\}\subset{\mathbb{C}}$

be a finite set of points in ${\mathbb{C}}$. Let $\Sigma^{\ast}$ be the set of
words with letters in $\Sigma$, with $e\in\Sigma^{\ast}$ being the empty word.
Let $|w|$ be the length of the word $w$ and $\widetilde{w}$ be $w$ in reversed
order. With the shuffle product

(7) $ua\,\hbox{\rus x}\,vb=(u\,\hbox{\rus x}\,vb)a+(ua\,\hbox{\rus
x}\,v)b,\quad\text{for }u,v\in\Sigma^{\ast},\;a,b\in\Sigma,$

and $e\,\hbox{\rus x}\,u=u=u\,\hbox{\rus x}\,e$ the ${\mathbb{Z}}$-span
$\langle\Sigma^{\ast}\rangle_{\mathbb{Z}}$ becomes a commutative ring.

For any word $w=a_{1}\ldots a_{n}\in\Sigma^{\ast}$ and any path
$\gamma:[0,1]\rightarrow{\mathbb{C}}\backslash\Sigma$ with $a_{0}=\gamma(0)$
and $a_{n+1}=\gamma(1)$ we define the iterated integral [12]

(8) $I(a_{0},a_{1}\ldots
a_{n},a_{n+1})=\int_{0<t_{1}<\ldots<t_{n}<1}\gamma^{\ast}\frac{\mathrm{d}t_{1}}{t_{1}-a_{1}}\wedge\ldots\wedge\gamma^{\ast}\frac{\mathrm{d}t_{n}}{t_{n}-a_{n}},$

where $\gamma^{\ast}\omega$ is the pullback of the differential form $\omega$
by $\gamma$.

Because (in this one-dimensional case) the integrand is closed, it is clear
that the iterated integral is a homotopy invariant. It neither depends on the
parametrization of the curve $\gamma$ nor on the shape of the path $\gamma$ as
long as the endpoints are fixed and no singularities are crossed. It is
convenient to generalize to the potentially singular case where the endpoints
$a_{0},a_{n+1}$ are in $\Sigma$. In this case the value of the iterated
integral may depend on the direction in which the endpoints are approached
(see, e.g., Section 2.2 in [17] for an elementary approach to iterated
integrals). In the case of hyperlogarithms (Section 3) the initial point is 0
and the endpoint is the variable $z$. One typically choses the initial point
to be approached from the positive real axis (this, e.g., gives $I(0,0,z)=\log
z$). An equivalent alternative is to shuffle-regularize iterated integrals,
see Section 3.2.

In the case of single-valued functions (Section 5) the regularization
convention for $\gamma$ is insignificant as long as it is used consistently.
In [19] the path $\gamma$ is always a straight line between the endpoints
(this gives $I(0,0,z)=\log|z|$).

In general, iterated integrals are very convenient to handle. An (incomplete)
list of identities (all of which are easy to verify) is [17]

I0:

$I(a_{0},a_{1})=1$ by definition.

I1:

$I(a_{0},w,a_{n+1})$ is independent of the parametrization of $\gamma$.

I2:

$I(a_{0},w,a_{n+1})$ is a homotopy invariant.

I3:

$I(a_{0},w,a_{0})=0$ for the constant path $\gamma=a_{0}$ and $|w|\geq 1$.

I4:

$I(a_{0},w,a_{n+1})=(-1)^{|w|}I(a_{n+1},\widetilde{w},a_{0})$ if the path
$\gamma$ is reversed.

I5:

For any $x\in{\mathbb{C}}$,

$I(a_{0},a_{1}\ldots a_{n},a_{n+1})=\sum_{k=0}^{n}I(a_{0},a_{1}\ldots
a_{k},x)I(x,a_{k+1}\ldots a_{n},a_{n+1})$

by path composition $\gamma=\gamma_{1}\gamma_{2}$ with
$\gamma_{1}(1)=x=\gamma_{2}(0)$.

I6:

For any $x\in{\mathbb{C}}$,

$I(a_{0},u,x)I(a_{0},v,x)=I(a_{0},u\,\hbox{\rus x}\,v,x)$

by shuffling the $t_{i}$ in the integration domain of (8).

I7:

For any $A\in{\mathbb{C}}^{\times}$, $B\in{\mathbb{C}}$ and $a_{0}\neq a_{1}$,
$a_{n}\neq a_{n+1}$,

$I(a_{0},a_{1}\ldots
a_{n},a_{n+1})=I(Aa_{0}+B,(Aa_{1}+B)\ldots(Aa_{n}+B),Aa_{n+1}+B)$

by substituting $t_{i}\mapsto(t_{i}-B)/A$ in (8).

I8:

For letters $a_{0},\ldots,a_{n+1}$ which depend on a variable $x$,

$\partial_{x}I(a_{0},a_{1}\ldots
a_{n},a_{n+1})=\sum_{k=1}^{n}\left(\partial_{x}\log\frac{a_{k+1}-a_{k}}{a_{k}-a_{k-1}}\right)I(a_{0},a_{1}\ldots
a_{k-1}a_{k+1}\ldots a_{n},a_{n+1})$

(where $\partial_{x}\log 0=0$) by using integration by parts in (8).

Note that in I7 the alphabet $\Sigma$ changes to $A\Sigma+B$. There exists an
analogous formula for any Möbius transformation $t\mapsto\frac{at+b}{ct+d}$
[17]. Singular iterated integrals are more subtle: The transformation
$t\mapsto t/A$ formally maps $I(0,0,1)=0$ to $I(0,0,A)=\log(A)$. If $A=1$,
however, I7 holds for all iterated integrals. Likewise, identities I0–I6 and
I8 also hold for singular iterated integrals.

There also exists a formula for the Galois coaction on iterated integrals [6,
15]. Although the existence of a coaction is central in the theory of iterated
integrals, we will not use it in this article.

##  3\. Hyperlogarithms

###  3.1. General theory

In this section we mostly follow the first sections of [5]. For completeness
we give (somewhat independent) proofs for the results in this section.

Consider the ring of regular functions on ${\mathbb{C}}\backslash\Sigma$,

(9) $\mathcal{O}_{\Sigma}={\mathbb{C}}[z,((z-a)^{-1})_{a\in\Sigma}].$

(One may think of the generator $z$ as being related to a puncture at
infinity.) For any word $w\in\Sigma^{\ast}$ we define the multivalued analytic
hyperlogarithm $L_{w}(z)$ as iterated integral,

(10) $L_{w}(z)=I(0^{+},w,z).$

The $L_{w}(z)$ span the space of hyperlogarithms over $\mathcal{O}_{\Sigma}$,

$\mathcal{H}\mathcal{L}_{\Sigma}=\langle
L_{w}(z),\;w\in\Sigma^{\ast}\rangle_{\mathcal{O}_{\Sigma}}.$

The homotopy dependence of $I(0^{+},w,z)$ reflects the multivaluedness of
$L_{w}(z)$. We get from I8

(11) $\partial_{z}L_{wa}(z)=\frac{1}{z-a}L_{w}(z)$

with (regualrized) limits $L_{w}(0)=0$. If $0\in\Sigma$, we get for a string
$0^{\\{n\\}}$ of $n$ letters 0

(12) $L_{0^{\\{n\\}}}(z)=\frac{1}{n!}\log^{n}(z).$

If the path of the iterated integral does not encircle singularities then
$L_{w}(z)$ can be expressed as multiple sum. With
$w=b_{1}0^{\\{n_{1}-1\\}}b_{2}0^{\\{n_{2}-1\\}}\ldots b_{r}0^{\\{n_{r}-1\\}}$
(i.e. $b_{1}=a_{1}$, $b_{2}=a_{n_{1}+1}$, …) we get

(13)
$L_{w}(z)=(-1)^{r}\sum_{0<k_{1}<k_{2}<\ldots<k_{r}}\frac{(\frac{b_{2}}{b_{1}})^{k_{1}}(\frac{b_{3}}{b_{2}})^{k_{2}}\cdots(\frac{z}{b_{r}})^{k_{r}}}{k_{1}^{n_{1}}k_{2}^{n_{2}}\cdots
k_{r}^{n_{r}}}\quad\text{for small }|z|.$

We refer to $|w|$ as the weight of the hyperlogarithm $L_{w}$. The number $r$
of non-zero letters in $w$ is the depth of $L_{w}$ (and of $w$). It will
follow from Theorem 4 that weight and depth of hyperlogarithms are well-
defined.

Let ${\mathbb{C}}\langle\Sigma\rangle$ be the ${\mathbb{C}}$-vector-space
freely generated by $\Sigma^{\ast}$. Following F. Brown in [4, 5] we use the
formal algebra (note that in a computer implementation one would use this
formal setup)

$\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}:={\mathbb{C}}\langle\Sigma\rangle\otimes_{\mathbb{C}}\mathcal{O}_{\Sigma}$

as model of hyperlogarithms. We consider
$\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$ as free $\mathcal{O}_{\Sigma}$
module where multiplication is on the right111In [5] multiplication is on the
left. Because of the connection to motivic Galois theory where the Galois
group conveniently coacts to the right we employ a left to right notation..

We consider hyperlogarithms as (analytic) realizations of the map

(14)
$\mathrm{eval}\,:\begin{array}[]{rcl}\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}&\rightarrow&\mathcal{H}\mathcal{L}_{\Sigma}\\\
w\otimes\phi(z)&\mapsto&\phi(z)L_{w}(z).\end{array}$

The $\mathcal{O}_{\Sigma}$ module $\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$
has a trivial Hopf-algebra structure with shuffle multiplication (7) and
deconcatenation as coproduct222The Galois coaction structure has no connection
to this Hopf-algebra structure.. The antipode is given by
$w\mapsto(-1)^{|w|}\widetilde{w}$ which, by I4, corresponds to path reversal.

By I6 eval maps the shuffle product to the pointwise product of complex
functions,

(15) $\mathrm{eval}\,[(u\,\hbox{\rus
x}\,v)\otimes\phi(z)]=\phi(z)L_{u}(z)L_{v}(z),\quad\text{for
}u,v\in\Sigma^{\ast},\;\phi\in\mathcal{O}_{\Sigma},$

which lifts eval to a homomorphism between the ${\mathbb{C}}$-algebras
$\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$ and
$\mathcal{H}\mathcal{L}_{\Sigma}$.

Words $0^{\\{k\\}}aw$, $a\neq 0$, $w\in\Sigma^{\ast}$, which begin in 0 have a
unique representation (see Lemma 3 in [20])

(16) $0^{\\{k\\}}aw=\sum_{i=0}^{k}(-1)^{i}0^{\\{k-i\\}}\,\hbox{\rus
x}\,a[0^{\\{i\\}}\,\hbox{\rus x}\,w].$

With (12), (13), and (15) this defines integral and sum representations for
hyperlogarithms of words which begin with 0,

(17) $L_{0^{\\{k\\}}aw}=\sum_{i=0}^{k}(-1)^{i}\frac{(\log
z)^{k-i}}{(k-i)!}L_{a[0^{\\{i\\}}\,\hbox{\rus x}\,w]}(z),$

where the hyperlogarithms are extended to $\Sigma^{\ast}$ by linearity. One
can use (12), (13), (17) or (11) and $L_{w}(0)=0$ as alternative definitions
of hyperlogarithms.

We make $\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$ a differential algebra by
defining a derivative $\partial_{z}$. For the empty word $e$ we set

$\partial_{z}(e\otimes\phi)=e\otimes\partial_{z}\phi(z).$

For all other words we define

(18)
$\partial_{z}(wa\otimes\phi)=wa\otimes\partial_{z}\phi(z)+w\otimes\frac{\phi(z)}{z-a}.$

For any $a\in\Sigma$ we define left and right derivatives—with respect to x ,
see (7)—$\delta_{a}^{\mathrm{l}}$ and $\delta_{a}^{\mathrm{r}}$ on
$\Sigma^{\ast}$ by

(19) $\delta_{a}^{\mathrm{l}}w=\left\\{\begin{array}[]{cc}u&\text{if }w=au,\\\
0&\text{otherwise,}\end{array}\right.\qquad\qquad\delta_{a}^{\mathrm{r}}w=\left\\{\begin{array}[]{cc}v&\text{if
}w=va,\\\ 0&\text{otherwise.}\end{array}\right.$

We get

(20)
$\partial_{z}=\text{id}\otimes\partial_{z}+\sum_{a\in\Sigma}\delta_{a}^{\mathrm{r}}\otimes\frac{1}{z-a}.$

The above formula implies that $\partial_{z}$ is a derivation on
$\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$. We find that eval is a
homomorphism of differential algebras.

We also define a residue $\mathrm{res}\,_{a}^{\mathcal{O}_{\Sigma}}$ for
$a\in{\mathbb{C}}$ acting on $\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$,

$\mathrm{res}\,_{a}^{\mathcal{O}_{\Sigma}}(w\otimes\phi)=w\otimes\mathrm{res}\,_{a}\phi,$

where $\mathrm{res}\,_{a}$ on the right hand side is the residue at $a$ in
$\mathcal{O}_{\Sigma}$.

Because $\mathrm{res}\,_{a}\partial_{z}=0$ we find that
$\mathrm{res}\,_{a}^{\mathcal{O}_{\Sigma}}\circ\partial_{z}$ decreases the
weights of the words in $\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$. The
residue in $\mathcal{H}\mathcal{L}_{\Sigma}$ needs a further evaluation at
$z=a$,

$\mathrm{res}\,_{a}=\mathrm{eval}\,_{a}\circ\mathrm{res}\,_{a}^{\mathcal{O}_{\Sigma}}:=\mathrm{eval}\,|_{z=a}\circ\mathrm{res}\,_{a}^{\mathcal{O}_{\Sigma}}.$

Without restriction we assume that any
$x=\sum_{w}w\otimes\phi_{w}\in\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$ is a
finite sum over distinct words $w$ with non-zero canceled fractions
$\phi_{w}$. Any such representation is unique. The maximum weight (length) of
words in $x$ induces a strict partial order on
$\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$. It is a standard technique in our
proofs to show that a minimal counter-example does not exist. To use this
technique efficiently it is convenient to refine the order induced by the
weight.

######  Definition 2.

Let $x=\sum_{w}w\otimes\phi_{w}\in\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$ as
above with weight $|x|=\max\\{|w|\\}$. Let $N_{x}=\\#\\{w,|w|=|x|\\}$ be the
number of words in the maximum weight part of $x$. The strict partial order
‘$\prec$’ on $\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$ is defined by

(21) $x\prec y\Leftrightarrow|x|<|y|\text{ or }(|x|=|y|\text{ and
}N_{x}<N_{y}).$

Note that $\partial_{z}f\not\succ f$ for all
$f\in\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$.

######  Lemma 3.

The kernel of $\partial_{z}$ in $\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$ is
$e\otimes{\mathbb{C}}$.

###### Proof.

Let $x\in\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$ be of minimum weight $n$
such that $\partial_{z}x=0$. Modulo lower weights $\partial_{z}$ only acts on
the right hand side of the tensor product. For $\partial_{z}x=0$ we hence need
that $\partial_{z}$ trivializes the fractions of the maximum weight part of
$x$. Hence, these fractions are constant. If the weight of $x$ is at least 1
then the maximum weight part of $x$ is of the form $\sum_{wa}wa\otimes c_{wa}$
with $0\neq c_{wa}\in{\mathbb{C}}$, $a\in\Sigma$, and $w\in\Sigma^{\ast}$ of
weight $n-1$. We get
$0=\mathrm{res}\,_{a}^{\mathcal{O}_{\Sigma}}\partial_{z}x=\sum_{w}w\otimes
c_{wa}$ modulo terms of weight $\leq n-2$. Hence all $c_{wa}=0$ and the claim
follows by contradiction. ∎

We get the following structure theorem for hyperlogarithms.

######  Theorem 4 (F. Brown 2004 [5]).

Let $\Sigma\subset{\mathbb{C}}$ be a finite set.

  1. (1)

The map eval is an isomorphism. In particular,
$\mathcal{H}\mathcal{L}_{\Sigma}$ is a free $\mathcal{O}_{\Sigma}$ module.

  2. (2)

The sequence

$0\longrightarrow{\mathbb{C}}\longrightarrow\mathcal{H}\mathcal{L}_{\Sigma}\stackrel{{\scriptstyle\partial_{z}}}{{\longrightarrow}}\mathcal{H}\mathcal{L}_{\Sigma}\longrightarrow
0$

is exact. I.e. the kernel of $\partial_{z}$ in
$\mathcal{H}\mathcal{L}_{\Sigma}$ is ${\mathbb{C}}$ and every
$f\in\mathcal{H}\mathcal{L}_{\Sigma}$ has a primitive
$F\in\mathcal{H}\mathcal{L}_{\Sigma}$ with $\partial_{z}F=f$.

  3. (3)

$\mathcal{H}\mathcal{L}_{\Sigma}$ is differentially simple. I.e. for every
$0\neq f\in\mathcal{H}\mathcal{L}_{\Sigma}$ there exists a differential
operator $D$ such that $Df=1$.

###### Proof.

We know that eval is a surjective differential homomorphism. To prove (1) we
show that there exists no $0\neq x\in\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$
with $\mathrm{eval}\,x=0$. Because the order $\prec$ is discrete we may assume
that $x$ is minimal with respect to $\prec$ and all $\Sigma$. Let
$w\otimes\phi$ be a term in $x$ of maximum weight. Because $\\{\phi=0\\}$ is a
finite set in ${\mathbb{C}}$ we may assume without restriction that
$\\{\phi=0\\}\subseteq\Sigma$. We have $\phi^{-1}\in\mathcal{O}_{\Sigma}$.
Because the term $w\otimes 1$ in $(e\otimes\phi^{-1})x$ loses weight upon
differentiation we have $\partial_{z}(e\otimes\phi^{-1})x\prec x$. Hence
$\partial_{z}(e\otimes\phi^{-1})x=0$ by minimality of $x$. From Lemma 3 we get
$(e\otimes\phi^{-1})x\in e\otimes{\mathbb{C}}$. Hence $x=e\otimes c\phi$ for
some $c\in{\mathbb{C}}$. From $\mathrm{eval}\,x=c\phi=0$ we get $c=0$ which is
a contradiction.

By statement (1) and Lemma 3 the kernel of $\partial_{z}$ is ${\mathbb{C}}$.
To prove (2) it hence suffices to show that every
$f\in\mathcal{H}\mathcal{L}_{\Sigma}$ has a primitive. We do this by induction
over the weight $n\in\\{-\infty,0,1,\ldots\\}$ of $f$. A primitive of $0$ is
0. For $f\neq 0$ we may assume by linearity using partial fraction
decomposition that $f(z)=(z-a)^{k}L_{w}(z)$ for $w\in\Sigma^{\ast}$,
$a\in\Sigma$, and $k\in{\mathbb{Z}}$. A primitive of $f(z)=(z-a)^{-1}L_{w}(z)$
is $L_{wa}(z)\in\mathcal{H}\mathcal{L}_{\Sigma}$. If $k\neq-1$, integration by
parts yields

$\int(z-a)^{k}L_{w}(z)\,\mathrm{d}z=\frac{(z-a)^{k+1}}{k+1}L_{w}(z)-\frac{1}{k+1}\int(z-a)^{k+1}\partial_{z}L_{w}(z)\,\mathrm{d}z.$

With (11) the right hand side exists in $\mathcal{H}\mathcal{L}_{\Sigma}$ by
induction.

We also prove (3) by induction over the weight $n$ of $f$. Let $0\neq
f\in\mathcal{H}\mathcal{L}_{\Sigma}$ and let $Q$ be the common denominator of
all coefficients of terms with weight $n$ in $f$. Then $Qf$ has only
polynomial coefficients in the maximum weight piece. Let $m$ be the maximum
degree of these polynomial. With $D_{0}=\partial_{z}^{m}Q$ we find that
$D_{0}f$ has weight $n$ but $\partial_{z}D_{0}f$ has weight $\leq n-1$. If
$D_{0}f=c\in{\mathbb{C}}^{\times}$ then $D=D_{0}/c$. Otherwise we obtain from
statement (2) that $\partial_{z}D_{0}f\neq 0$. By induction there exists a
differential operator $D_{1}$ such that $D=D_{1}\partial_{z}D_{0}$ fulfills
$Df=1$. ∎

###  3.2. Regularized evaluation

With the definition

(22) $L_{0^{\\{n\\}}}(0)=0,\quad\text{if }n\geq 1,$

we get $L_{w}(0)=0$ for all $w\neq e$.

The evaluation of a hyperlogarithm $L_{w}(z)$ at $z=a\neq 0$ is singular if
$w$ ends in $a$. In this case we can use (16) from the right (swap the order
of all letters) to shuffle-regularize $w$. With the map $t\mapsto a-t$ in (8)
(see I7) and path reversal I4 it is natural to define

(23) $L_{a^{\\{n\\}}}(a)=(-1)^{n}L_{0^{\\{n\\}}}(a).$

Identities (22) and (23) regularize the evaluation of hyperlogarithms
$L_{w}(a)$ for any word $w$ at any $a\in\Sigma$.

Note that the prescription is a definition. Had we used the map $t\mapsto t-a$
in I7, the right hand side of (23) would have been
$(-1)^{n}L_{0^{\\{n\\}}}(-a)$ (which possibly generates a new letter $-a$). In
the case of single-valued functions where $\log|a|$ replaces $\log(a)$ both
prescriptions are equivalent. We skip a detailed treatment of regularization
here and use (23) as definition.

######  Example 5.

We want to evaluate $L_{012}(z)$ at $z=2$. Using (16), (17), and (23) we get

$\displaystyle L_{012}(2)$ $\displaystyle=$ $\displaystyle
L_{2}(2)L_{01}(2)-L_{201}(2)-L_{021}(2)$ $\displaystyle=$ $\displaystyle-
L_{0}(2)[L_{0}(2)L_{1}(2)-L_{10}(2)]-L_{201}(2)-L_{0}(2)L_{21}(2)+L_{201}(2)+L_{210}(2)$
$\displaystyle=$ $\displaystyle-(\log 2)^{2}L_{1}(2)+\log
2[L_{10}(2)-L_{21}(2)]+L_{210}(2).$

###  3.3. Series expansions of hyperlogarithms

Let

$\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}=\mathrm{eval}\,\Sigma^{\ast}\otimes{\mathbb{C}}$

be the ${\mathbb{C}}$-algebra of hyperlogarithms with constant coefficients.

With (13) we obtain from (17) an expansion of $L_{w}(z)$ for any word $w$,

(24) $L_{w}(z)=\sum_{\ell=0}^{|w|}\sum_{m=0}^{\infty}c_{\ell,m}^{0}(\log
z)^{\ell}z^{m},\quad c_{\ell,m}^{0}\in{\mathbb{C}},$

which converges in a neighborhood of 0. This generalizes to all values
$a\in\Sigma$.

######  Lemma 6.

Let $f\in\mathcal{H}\mathcal{L}_{\Sigma}$ have weight $n$. For any
$a\in\Sigma$ there exists an $M_{a}\in{\mathbb{Z}}$ such that

(25)
$f(z)=\sum_{\ell=0}^{n}\sum_{m=M_{a}}^{\infty}c_{\ell,m}^{a}[\log(z-a)]^{\ell}(z-a)^{m},\quad
c_{\ell,m}^{a}\in{\mathbb{C}},$

in a neighborhood of $z=a$. If
$f\in\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$ then $M_{a}=0$.

###### Proof.

For $f\in\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$ the result follows from
(24) by I7 with $A=1,B=-a$ and path concatenation I5 at $x=0$. To obtain the
expansion for a general $f\in\mathcal{H}\mathcal{L}_{\Sigma}$ one has to
Laurent expand the coefficients in $\mathcal{O}_{\Sigma}$. In this case
$M_{a}$ is the maximum pole order of the coefficients at $z=a$. ∎

Note that the values of the coefficients $c_{\ell,m}^{a}$ depend on the sheet
on which the multivalued hyperlogarithm $f$ is expanded (see next subsection).

At $z=\infty$ there exists an analogous expansion in $1/z$ (use Section 4.2
and (24) for $f(1/z)$),

(26)
$f(z)=\sum_{\ell=0}^{n}\sum_{m=-\infty}^{M_{\infty}}c_{\ell,m}^{\infty}(\log
z)^{\ell}z^{m},\quad c_{\ell,m}^{\infty}\in{\mathbb{C}},$

with $M_{\infty}=0$ if $f\in\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$. For
$a\in{\mathbb{C}}\setminus\Sigma$ the function
$f\in\mathcal{H}\mathcal{L}_{\Sigma}$ is holomorphic at $z=a$.

###  3.4. Monodromy

From now on we assume that the path of integration in $L_{w}(z)$ is a straight
line from $0^{+}$ to $z$ with regularizations (22) and (23) if necessary.

For $z\in{\mathbb{C}}\backslash\Sigma$ fix a cycle basis
$\\{\gamma_{a},a\in\Sigma\\}$ of the homotopy group of
${\mathbb{C}}\backslash\Sigma$ where a cycle
$\gamma_{a}=\gamma_{za}\gamma_{aa}\gamma_{za}^{-1}\subset{\mathbb{C}}\backslash\Sigma$
goes from $z$ to a regular neighborhood of $a$ (by $\gamma_{za}$), encircles
$a$ in counter-clockwise direction (by $\gamma_{aa}$), and goes back to $z$
following $\gamma_{za}$ with reversed orientation. Define

(27) $\mathcal{M}_{a}L_{w}(z)=\gamma_{a}^{\ast}L_{w}(z),$

which means we analytically continue $L_{w}(z)$ from $z$ back to $z$ along
$\gamma_{a}$. Note that, by construction, $\mathcal{M}_{a}$ is an algebra
homomorphism. The operators $\mathcal{M}_{a}$ generate a representation of the
homotopy group of ${\mathbb{C}}\backslash\Sigma$. A change of the cycle basis
amounts a conjugation of the generators $\mathcal{M}_{a}$.

Let us first study $\mathcal{M}_{0}$. For $\gamma_{z0}$ we use the straight
line from $z$ to a small positive $\epsilon$. We assume that there are no
singularities on this path (otherwise we have to move $z$ slightly). We get
$\gamma_{0}^{\ast}L_{0}(z)=\gamma_{0}^{\ast}\log(z)=L_{0}(z)+2\pi\mathrm{i}$.
With (12) we find

(28)
$\mathcal{M}_{0}L_{0^{\\{n\\}}}(z)=\sum_{j=0}^{n}\frac{(2\pi\mathrm{i})^{j}}{j!}L_{0^{\\{n-j\\}}}(z).$

With (13) it is clear that $\mathcal{M}_{0}$ is trivial on $L_{w}(z)$ for all
words $w$ that do not begin in 0.

######  Lemma 7.

For any word $0^{\\{k\\}}w$, $w\in\Sigma^{\ast}$ with
$\delta_{0}^{\mathrm{l}}w=0$, Equation (28) generalizes to (Lemma 2.2 in [17])

(29)
$\mathcal{M}_{0}L_{0^{\\{k\\}}w}(z)=\sum_{j=0}^{k}\frac{(2\pi\mathrm{i})^{j}}{j!}L_{0^{\\{k-j\\}}w}(z).$

If we pull $\mathcal{M}_{0}$ back to
$\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$ using the isomorphism eval we get

(30)
$\mathcal{M}_{0}0^{\\{k\\}}w\otimes\phi=\sum_{j=0}^{k}0^{\\{k-j\\}}w\otimes\frac{(2\pi\mathrm{i})^{j}\phi}{j!}.$

###### Proof.

With (28) it suffices to show (30) for $w=au$ with $0\neq a\in\Sigma$. From
(16) and (28) we get for the shuffle homomorphism $\mathcal{M}_{0}$,

$\mathcal{M}_{0}0^{\\{k\\}}au\otimes\phi=\mathcal{M}_{0}\sum_{i=0}^{k}(-1)^{i}0^{\\{k-i\\}}\,\hbox{\rus
x}\,a[0^{\\{i\\}}\,\hbox{\rus
x}\,w]\otimes\phi=\sum_{i=0}^{k}\sum_{j=0}^{k-i}(-1)^{i}0^{\\{k-i-j\\}}\,\hbox{\rus
x}\,a[0^{\\{i\\}}\,\hbox{\rus
x}\,w]\otimes\frac{(2\pi\mathrm{i})^{j}\phi}{j!}.$

We swap the sums and use (16) again to evaluate the sum over $i$ yielding
(30). ∎

It is convenient to write $\mathcal{M}_{0}$ as an exponential,

(31) $\mathcal{M}_{0}=\exp(2\pi\mathrm{i}m_{0}),\quad\text{with
}m_{0}=\delta_{0}^{\mathrm{l}}\otimes 1.$

The infinitesimal monodromy $m_{0}$ is a nilpotent derivation on
$\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$. From (31) it is clear that
$\mathcal{M}_{0}$ is unipotent.

At $z=0$ the right hand side of (29) vanishes if $w\neq e$. We get for the
regularized limit

(32)
$\mathrm{eval}\,_{0}\mathcal{M}_{0}L_{w}(z)=\left\\{\begin{array}[]{cl}\frac{(2\pi\mathrm{i})^{n}}{n!}&\text{if
}w=0^{\\{n\\}},\\\ 0&\text{otherwise.}\end{array}\right.$

For general $a$ we choose the path $\gamma_{a}=\gamma_{\epsilon
z}^{-1}\gamma_{\epsilon a}\gamma_{aa}\gamma_{\epsilon a}^{-1}\gamma_{\epsilon
z}$ to have five sections: From $z$ straight to $\epsilon$, from $\epsilon$
straight to $a$, a tiny circle around $a$ in counter-clockwise direction, from
$a$ straight back to $\epsilon$, and from $\epsilon$ straight back to $z$. We
assume that $\epsilon>0$ is small and that there are no singularities on this
path.

In the limit $\epsilon\to 0^{+}$ the first part of $\gamma_{a}$ cancels the
path $\gamma_{0z}$ from $0^{+}$ to $z$ of $L_{w}(z)$. By path concatenation we
get
$\gamma_{0z}\gamma_{a}\to\gamma_{0a}\gamma_{aa}\gamma_{0a}^{-1}\gamma_{0z}$.
Translating (32) to $z=a$ the local monodromy operator $\gamma_{aa}$ maps
sequences $a^{\\{k\\}}$ to $(2\pi\mathrm{i})^{k}/k!$ and is zero otherwise.
With I4 and I5 we obtain

(33) $\mathcal{M}_{a}(w\otimes 1)=\sum_{k\geq
0}\sum_{w=ua^{\\{k\\}}vx}\frac{(-1)^{|v|}(2\pi\mathrm{i})^{k}}{k!}L_{u}(a)L_{\widetilde{v}}(a)\,x\otimes
1$

for words $u,v,x\in\Sigma^{\ast}$. The infinitesimal monodromy $m_{a}$ is the
$2\pi\mathrm{i}$-coefficient,

(34) $\mathcal{M}_{a}=\exp(2\pi\mathrm{i}m_{a}),\quad\text{with}\quad
m_{a}(w\otimes
1)=\sum_{w=uavx}(-1)^{|v|}L_{u}(a)L_{\widetilde{v}}(a)\,x\otimes 1.$

If $w$ contains a sequence of at least two letters $a$ then (33) and (34) have
singular terms which are defined by regularitzation, see Section 3.2. One can
use the reversed version of (16) to derive a fully regular formula for the
monodromy of a regular word (i.e. a word that does not begin in 0). So, the
monodromy of regular words does not depend on the regularization prescription.

To see that $\mathcal{M}_{a}=\exp(2\pi\mathrm{i}m_{a})$ one has to use that
$\sum_{w=vu}(-1)^{|v|}L_{\widetilde{v}}(a)L_{u}(a)=\delta_{w,e}$ by path-
reversal I4, path composition I5, and the triviality of the constant path I3
($\delta_{a,b}=1$ if $a=b$ and zero otherwise). The sum in $m_{a}^{k}$
collapses from $w=u_{1}av_{1}u_{2}av_{2}\ldots u_{k}av_{k}x$ to
$w=u_{1}a^{\\{k\\}}v_{k}x$.

Likewise, we see that the evaluation at $z=a$ cuts off a letter $a$ from the
right,

(35)
$\mathrm{eval}\,_{a}m_{a}=\mathrm{eval}\,_{a}\delta_{a}^{\mathrm{r}}\otimes
1.$

With regularization (22) it is also clear that (34) reduces to (31) in the
case $a=0$.

We see that $\partial_{z}$ and $m_{a}$ commute,

(36) $[\partial_{z},m_{a}]=0.$

With eval we lift (34) to a formula in $\mathcal{H}\mathcal{L}_{\Sigma}$,

(37)
$m_{a}L_{w}(z)=\sum_{w=uavx}(-1)^{|v|}L_{u}(a)L_{\widetilde{v}}(a)L_{x}(z).$

Modulo lower weights we have $m_{a}=\delta_{a}^{\mathrm{l}}\otimes 1$, i.e.

(38) $m_{a}L_{bw}(z)=\delta_{a,b}L_{w}(z)+\text{terms of lower weight.}$

For later use we rewrite (35) using the residue operator on
$\mathcal{H}\mathcal{L}_{\Sigma}$,

(39) $\mathrm{eval}\,_{a}m_{a}=\mathrm{res}\,_{a}\partial_{z}.$

###  3.5. A commutative hexagon

We have defined two derivations on $\mathcal{H}\mathcal{L}_{\Sigma}$, namely
$\partial_{z}$ and $m_{a}$. Because hyperlogarithms are analytic the
derivative $\partial_{\overline{z}}$ with respect to the complex conjugate of
$z$ is zero. Later, when we define GSVHs, we sacrifice analyticity for single-
valuedness. This means that $m_{a}$ becomes trivial whereas
$\partial_{\overline{z}}$ becomes non-trivial. The main tool for the
construction of GSVHs will be a commutative hexagon which has a (simpler)
analogue in the analytic case with $\partial_{\overline{z}}$ replaced by
$m_{a}$. We present this analogue as introduction to GSVHs. An application of
the commutative hexagon for hyperlogarithms is integration in the motivic
$f$-alphabet [23].

Before we prove commutativity of the hexagon we need to define a generic
infinitesimal monodromy (covering all points $a\in\Sigma$). To this end we
extend $\mathcal{H}\mathcal{L}_{\Sigma}$ to
$\Sigma^{\ast}\otimes\mathcal{H}\mathcal{L}_{\Sigma}$ where the first factor
keeps track of the puncture $a$. By $f\mapsto e\otimes f$ we embed
$\mathcal{H}\mathcal{L}_{\Sigma}$ into
$\Sigma^{\ast}\otimes\mathcal{H}\mathcal{L}_{\Sigma}$. (A model for
$\Sigma^{\ast}\otimes\mathcal{H}\mathcal{L}_{\Sigma}$ is
$\Sigma^{\ast}\otimes\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$ which we do not
use here.) We define

(40)
$m:\begin{array}[]{rcl}\Sigma^{\ast}\otimes\mathcal{H}\mathcal{L}_{\Sigma}&\longrightarrow&\Sigma^{\ast}\otimes\mathcal{H}\mathcal{L}_{\Sigma}\\\
w\otimes f(z)&\mapsto&\sum_{a\in\Sigma}wa\otimes m_{a}f(z).\end{array}$

The derivative $\partial_{z}$ only acts on the right factor of
$\Sigma^{\ast}\otimes\mathcal{H}\mathcal{L}_{\Sigma}$.

######  Proposition 8.

Let $\Sigma^{\ast}_{>}=\Sigma^{\ast}\backslash\\{e\\}$ be the set of non-empty
words. The sequence

$0\longrightarrow\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}\longrightarrow\Sigma^{\ast}\otimes\mathcal{H}\mathcal{L}_{\Sigma}\stackrel{{\scriptstyle
m}}{{\longrightarrow}}\Sigma^{\ast}_{>}\otimes\mathcal{H}\mathcal{L}_{\Sigma}\longrightarrow
0$

is exact. I.e. the kernel of $m$ in
$\Sigma^{\ast}\otimes\mathcal{H}\mathcal{L}_{\Sigma}$ is
$\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$ and for every
$x\in\Sigma^{\ast}_{>}\otimes\mathcal{H}\mathcal{L}_{\Sigma}$ there exists an
$X\in\Sigma^{\ast}\otimes\mathcal{H}\mathcal{L}_{\Sigma}$ such that $mX=x$.

###### Proof.

We first show that the kernel of $m$ is
$\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}$. It is clear that
$m\,\Sigma^{\ast}\otimes\mathcal{O}_{\Sigma}=0$. Assume that there exists an
$x\in\Sigma^{\ast}\otimes\mathcal{H}\mathcal{L}_{\Sigma}$ with $mx=0$ and
weight $n\geq 1$ (in the right factor). We uniquely write
$x=\sum_{a\in\Sigma,u,v\in\Sigma^{\ast}}u\otimes\phi_{uav}(z)L_{av}(z)$ with
$0\neq\phi_{uav}\in\mathcal{O}_{\Sigma}$. By (38) we have
$0=mx=\sum_{a,u,v}ua\otimes\phi_{uav}(z)L_{v}(z)$ modulo weights $\leq n-2$.
By Theorem 4 (1) we get $\sum_{a,u}ua\otimes\phi_{uav}(z)=0$ for all $v$ of
weight $n-1$. Hence $\phi_{uav}(z)=0$ which is a contradiction.

We prove the existence of $X$ by induction over the weight $n$ of
$x\in\Sigma^{\ast}_{>}\otimes\mathcal{H}\mathcal{L}_{\Sigma}$. For $x=0$ we
set $X=0$. For $n\geq 0$ we use linearity to assume that
$x=ua\otimes\phi(z)L_{v}(z)$ with
$a\in\Sigma,u,v\in\Sigma^{\ast},\phi\in\mathcal{O}_{\Sigma},L_{v}\in\mathcal{H}\mathcal{L}_{\Sigma}$.
We set $X=u\otimes\phi(z)L_{av}(z)-Y$ and get from (38) that $mX=x+y-mY$ with
$y$ of weight $\leq n-1$. By induction there exists an $Y$ such that $mY=y$
which completes the proof. ∎

Consider the ${\mathbb{C}}$-algebra of hyperlogarithms with constant
coefficients $\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$. The derivative
$\partial_{z}$ maps $\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$ onto
$\partial_{z}\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$, the
${\mathbb{C}}$-vector-space of hyperlogarithms with simple pole coefficients.
Likewise $m$ maps $\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$ to
$m\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}=\Sigma\otimes\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$,
the ${\mathbb{C}}$-vector-space of hyperlogarithms with single letters as left
factors. We also define
$\partial_{z}m\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}=m\partial_{z}\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}=\Sigma\otimes\partial_{z}\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$,
see (36).

On $\mathcal{H}\mathcal{L}_{\Sigma}$ we have the residue
$\mathrm{res}\,_{a}:\mathcal{H}\mathcal{L}_{\Sigma}\rightarrow{\mathbb{C}}$.
We define the projection $\pi_{\partial_{z}}$ onto the residue-free part of
$\mathcal{H}\mathcal{L}_{\Sigma}$,

(41)
$\pi_{\partial_{z}}:\begin{array}[]{rcl}\mathcal{H}\mathcal{L}_{\Sigma}&\rightarrow&\mathcal{H}\mathcal{L}_{\Sigma}\\\
f(z)&\mapsto&f(z)-\sum_{a\in\Sigma}\frac{\mathrm{res}\,_{a}f}{z-a}\end{array}$

with trivial extension to
$\Sigma^{\ast}\otimes\mathcal{H}\mathcal{L}_{\Sigma}$. The projection
$\pi_{m}$ is the identity on $e\otimes\mathcal{H}\mathcal{L}_{\Sigma}$. On
$\Sigma^{\ast}_{>}\otimes\mathcal{H}\mathcal{L}_{\Sigma}$ we define

(42)
$\pi_{m}:\begin{array}[]{rcl}\Sigma^{\ast}_{>}\otimes\mathcal{H}\mathcal{L}_{\Sigma}&\rightarrow&\Sigma^{\ast}_{>}\otimes\mathcal{H}\mathcal{L}_{\Sigma}\\\
wa\otimes f(z)&\mapsto&wa\otimes(f(z)-f(a)).\end{array}$

We have the obvious identities

(43)
$m\pi_{\partial_{z}}=m\quad\text{and}\quad\partial_{z}\pi_{m}=\partial_{z}.$

Finally, we define on
$\partial_{z}\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$ and on
$m\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$ an integral and an inverse of
$m$ (respectively) which maps into the subspace of
$\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$ which evaluates to 0 at $z=0$
(possibly after regularization, Section 3.2),

$\displaystyle\int_{0}\mathrm{d}z:\partial_{z}\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}\rightarrow\mathcal{H}\mathcal{L}_{\Sigma},$
$\displaystyle f\mapsto F(z)-F(0)\quad\text{if}\quad\partial_{z}F=f,$ (44)
$\displaystyle
m^{-1}:m\mathcal{H}\mathcal{L}_{\Sigma}\rightarrow\mathcal{H}\mathcal{L}_{\Sigma},$
$\displaystyle x\mapsto X(z)-X(0)\quad\text{if}\quad mX=x.$

The operators $\int_{0}\mathrm{d}z$ and $m^{-1}$ exist and are well-defined by
Theorem 4 (2) and by Proposition 8. The integral $\int_{0}\mathrm{d}z$ acts on
the right factor of
$\Sigma^{\ast}\otimes\partial_{z}\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$.

1.0 Black
[arrow,arrowpos=1,arrowlength=5,arrowwidth=2,arrowinset=0.2](48,38)(48,56)
[arrow,arrowpos=1,arrowlength=5,arrowwidth=2,arrowinset=0.2](128,38)(128,56)
[arrow,arrowpos=1,arrowlength=5,arrowwidth=2,arrowinset=0.2](54,17)(71,7)
[arrow,arrowpos=1,arrowlength=5,arrowwidth=2,arrowinset=0.2](122,77)(105,87)
[arrow,arrowpos=1,arrowlength=5,arrowwidth=2,arrowinset=0.2](105,7)(122,17)
[arrow,arrowpos=1,arrowlength=5,arrowwidth=2,arrowinset=0.2](54,77)(71,87)
(42,24)[lb]$\partial_{z}\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$(122,24)[lb]$m\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$(122,65)[lb]$m\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$(42,65)[lb]$\partial_{z}\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$(32,43)[lb]$\pi_{\partial_{z}}$(133,43)[lb]$\pi_{m}$(40,84)[lb]$\int_{0}\mathrm{d}z$(118,84)[lb]$m^{-1}$(69,-5)[lb]$m\partial_{z}\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$(78,88)[lb]$\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$(52,2)[lb]$m$(117,0)[lb]$\int_{0}\mathrm{d}z$

Figure 2. The inductive construction of hyperlogarithms with a commutative
hexagon.

######  Theorem 9.

The diagram in Figure 2 commutes.

###### Proof.

Let $f\in\partial_{z}\mathcal{H}\mathcal{L}_{\Sigma}^{\mathbb{C}}$ and
$f_{1}=\int_{0}\pi_{\partial_{z}}f\mathrm{d}z$,
$f_{2}=m^{-1}\pi_{m}\int_{0}mf\mathrm{d}z$. We need to show that
$f_{1}=f_{2}$.

By (36) and (43) we have
$\partial_{z}m(f_{1}-f_{2})=m\pi_{\partial_{z}}f-\partial_{z}\pi_{m}\int_{0}mf\mathrm{d}z=mf-
mf=0$. By Theorem 4 (2) we get $m(f_{1}-f_{2})\in\Sigma\otimes{\mathbb{C}}$.
From (38) and Proposition 8 we get

$f_{1}(z)-f_{2}(z)=\sum_{a\in\Sigma}c_{a}L_{a}(z)+d\quad\text{with
}c_{a},d\in{\mathbb{C}}.$

Because $f_{1}(0)=f_{2}(0)=0$ we have $d=0$. We choose $b\in\Sigma$ and apply
$\mathrm{eval}\,_{b}m_{b}$ to the above equation,

$[m_{b}(f_{1}-f_{2})](b)=c_{b}.$

By (39) we have $(m_{b}f_{1})(b)=\mathrm{res}\,_{b}\partial_{z}f_{1}=0$
because $\partial_{z}f_{1}=\pi_{\partial_{z}}f$ is in the image of
$\pi_{\partial_{z}}$. Likewise $(m_{b}f_{2})(b)=0$ because $mf_{2}$ is in the
image of $\pi_{m}$. We find $c_{b}=0$ for all $b\in\Sigma$ and hence
$f_{1}=f_{2}$. ∎

The following examples illustrate the role of $\pi_{\partial_{z}}$ and
$\pi_{m}$.

######  Example 10.

Let $f(z)=z^{-1}\in\partial_{z}\mathcal{H}\mathcal{L}_{\\{0\\}}^{\mathbb{C}}$.
Then $\int_{0}f(z)\mathrm{d}z=L_{0}(z)=\log(z)$ whereas $mf(z)=0$. The hexagon
commutes because $\pi_{\partial_{z}}z^{-1}=0$.

######  Example 11.

Let
$f(z)=L_{a}(z)/(z-b)\in\partial_{z}\mathcal{H}\mathcal{L}_{\\{a,b\\}}^{\mathbb{C}}$
for $a\neq b\in{\mathbb{C}}$. Then
$\pi_{\partial_{z}}f(z)=[L_{a}(z)-L_{a}(b)]/(z-b)$ and

$\int_{0}\pi_{\partial_{z}}f(z)\mathrm{d}z=L_{ab}(z)-L_{a}(b)L_{b}(z)\in\mathcal{H}\mathcal{L}_{\\{a,b\\}}^{\mathbb{C}}.$

On the other hand we have $mf(z)=a\otimes(z-b)^{-1}$ and
$\int_{0}mf(z)\mathrm{d}z=a\otimes L_{b}(z)$. With (37), we get

$mL_{ab}(z)=a\otimes L_{b}(z)-a\otimes L_{b}(a)+b\otimes L_{a}(b).$

Therefore

$m^{-1}a\otimes L_{b}(z)=L_{ab}(z)+L_{b}(a)L_{a}(z)-L_{a}(b)L_{b}(z).$

Direct application of $m^{-1}$ to $a\otimes L_{b}(z)$ does not reproduce
$L_{ab}(z)-L_{a}(b)L_{b}(z)$. With $\pi_{m}a\otimes L_{b}(z)=a\otimes
L_{b}(z)-a\otimes L_{b}(a)$ and $m^{-1}a\otimes L_{b}(a)=L_{b}(a)L_{a}(z)$ the
spurious term $L_{b}(a)L_{a}(z)$ cancels.

##  4\. Generalized hyperlogarithms

Let ${\mathbb{F}}=\overline{{\mathbb{F}}}\subseteq{\mathbb{C}}$ be a
quadratically closed number field, i.e.

(45)
$x\in{\mathbb{F}}\subseteq{\mathbb{C}}\Rightarrow\overline{x},\pm\sqrt{x}\in{\mathbb{F}}.$

The smallest field ${\mathbb{F}}$ are the constructible numbers, the largest
field is ${\mathbb{F}}={\mathbb{C}}$.

######  Remark 12.

For applications in massless pQFT the fields ${\mathbb{Q}}(\mathrm{i})$ and
${\mathbb{Q}}(\mathrm{e}^{\pi\mathrm{i}/3})$ possibly suffice [10, 20, 21]. We
expect that pQFT uses only a small subspace of the GSVHs defined in Section 6.

To describe single-valued hyperlogarithms on the punctured complex plane we
need to give up analyticity. Because the differentials $\partial_{z}$ and
$\partial_{\overline{z}}$ commute, in most situations one has the option to
either consider $z$ and ${\overline{z}}$ as complex conjugates or as
independent variables. Depending on the context we shift from one picture to
the other. We use double arguments $z,{\overline{z}}$ in functions if we
consider $z$ and ${\overline{z}}$ as independent variables.

######  Remark 13.

With minimal modifications it is possible to develop a very similar theory of
GSVHs where $z$ and ${\overline{z}}$ are replaced by any linear combination of
$z$ and ${\overline{z}}$ (or more general functions of $z,{\overline{z}}$).
One may, e.g., use real and imaginary parts of $z$. For the (rather special)
Gegenbauer method in [1, 17] one needs to use the modulus and the argument of
$z$. In many situations, however, the variables $z$ and ${\overline{z}}$ are
most efficient. In pQFT, e.g., one has to invert the Laplacian
$\partial_{z}\partial_{\overline{z}}$ which factorizes in $z$ and
${\overline{z}}$.

In contrast to the concept of single-valuedness (Section 5), the definition of
generalized hyperlogarithms is of technical nature. In pQFT there exists no
fundamental property that requires generalized hyperlogarithms. They are
merely the most general functions that are well understood and can be handled
efficiently. For small graphs most graphical functions can be expressed in
terms of generalized hyperlogarithms [1, 17].

###  4.1. Fractional linear transformations

Let

(46) $\mathrm{FLT}\,_{\mathbb{F}}=\Big{\\{}\beta(z)=\frac{az+b}{cz+d},\quad
a,b,c,d\in{\mathbb{F}}\Big{\\}}$

be the set of fractional linear transformations (FLTs) with coefficients in
${\mathbb{F}}$. The FLT $\beta$ is invertible if $ad-bc\neq 0$. Then $\beta$
is an automorphism of ${\mathbb{C}}\cup\\{\infty\\}$ (a Möbius
transformation). Because

$\partial_{z}\frac{az+b}{cz+d}=\frac{ad-bc}{(cz+d)^{2}},$

$ad-bc=0$ is equivalent to constant $\beta$. For any $\beta(z)=(az+b)/(cz+d)$
with $ad-bc\neq 0$ the inverse of $\beta$ is (we do not use $\beta^{-1}$ for
the reciprocal of $\beta$)

$\beta^{-1}(z)=\frac{dz-b}{-cz+a}.$

######  Lemma 14.

For any variables $z,{\overline{z}}$ and
$\beta(z)=(az+b)/(cz+d)\in\mathrm{FLT}\,_{\mathbb{F}}$ with $ad-bc\neq 0$ we
have

$\partial_{z}\log({\overline{z}}-\beta(z))=\frac{1}{z-\beta^{-1}({\overline{z}})}-\frac{1}{z+d/c},$

where the second term on the right hand side is absent if $c=0$.

###### Proof.

Because $ad-bc\neq 0$ either $a\neq 0$ or $c\neq 0$. We get

${\overline{z}}-\beta(z)=\frac{(c{\overline{z}}-a)(z-\beta^{-1}({\overline{z}}))}{cz+d}.$

Taking $\partial_{z}\log$ gives the result. ∎

In general, the product of two FLTs is not a sum of FLTs (it may have
squares). Here, we need the converse property that the difference of two FLTs
is the product of two FLTs.

######  Lemma 15.

Let $\beta_{1},\beta_{2}\in\mathrm{FLT}\,_{\mathbb{F}}$.

  1. (1)

There exist $\beta_{3},\beta_{4}\in\mathrm{FLT}\,_{\mathbb{F}}$ such that

$\beta_{1}(z)-\beta_{2}(z)=\beta_{3}(z)\cdot\beta_{4}(z)$

  2. (2)

We have

$\det\left(\begin{array}[]{cc}\partial_{z}\log(z-\beta_{1}({\overline{z}}))&\partial_{\overline{z}}\log(z-\beta_{1}({\overline{z}}))\\\
\partial_{z}\log(z-\beta_{2}({\overline{z}}))&\partial_{\overline{z}}\log(z-\beta_{2}({\overline{z}}))\end{array}\right)=[\partial_{\overline{z}}\log(\beta_{1}({\overline{z}})-\beta_{2}({\overline{z}}))]\left[\partial_{z}\log\frac{z-\beta_{1}({\overline{z}})}{z-\beta_{2}({\overline{z}})}\right].$

###### Proof.

For $a_{1},b_{1},c_{1},d_{1},a_{2},b_{2},c_{2},d_{2}\in{\mathbb{F}}$ we have

$\frac{a_{1}z+b_{1}}{c_{1}z+d_{1}}-\frac{a_{2}z+b_{2}}{c_{2}z+d_{2}}=\frac{(a_{1}c_{2}-a_{2}c_{1})z^{2}+(a_{1}d_{2}-a_{2}d_{1}+b_{1}c_{2}-b_{2}c_{1})z+b_{1}d_{2}-b_{2}d_{1}}{(c_{1}z+d_{1})(c_{2}z+d_{2})}.$

Because ${\mathbb{F}}$ is quadratically closed, the numerator factorizes in
${\mathbb{F}}$ and the product may be expressed as $\beta_{3}(z)\beta_{4}(z)$
for a suitable choice of $\beta_{3}(z)$ and $\beta_{4}(z)$. Statement (2) is
an explicit calculation. ∎

Statement (1) is the sole reason for our restriction to quadratically closed
fields.

###  4.2. Iterated integrals with parameters

Let

$\mathcal{H}\mathcal{L}_{\mathbb{F}}=\bigcup_{\Sigma\subset{\mathbb{F}}}\mathcal{H}\mathcal{L}_{\Sigma}.$

Likewise $\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}$ is the
${\mathbb{C}}$-algebra of hyperlogarithms in
$\mathcal{H}\mathcal{L}_{\mathbb{F}}$ with constant coefficients in
${\mathbb{C}}$.

Any iterated integral with letters in ${\mathbb{C}}(z)$ is in
$\mathcal{H}\mathcal{L}_{\mathbb{C}}^{\mathbb{C}}$. A bootstrap algorithm for
the conversion to constant letters is given by differentiating the iterated
integral with respect to $z$ using I8. This lowers the weight of the iterated
integral. By induction the derivative is in
$\mathcal{H}\mathcal{L}_{\mathbb{C}}^{\mathbb{C}}$. A priori, the primitive is
in $\mathcal{H}\mathcal{L}_{\mathbb{C}}$. Using I8 it is not hard to see that
the primitive is in $\mathcal{H}\mathcal{L}_{\mathbb{C}}^{\mathbb{C}}$. This
algorithm was suggested by F. Brown [2, 3] and later implemented by E. Panzer
[16].

To emphasize the dependence on the variable $z$ (or ${\overline{z}}$) we will
give $\mathcal{H}\mathcal{L}_{\mathbb{F}}$ and $\mathrm{FLT}\,_{\mathbb{F}}$
an argument. So, $\mathcal{H}\mathcal{L}_{\Sigma}(z)$ denotes the
${\mathbb{C}}$-algebra of hyperlogarithms in $z$ and
$\mathrm{FLT}\,_{\mathbb{F}}(z)$ is the set of FLTs over ${\mathbb{F}}$ in
$z$.

######  Lemma 16.

We consider $z$ and ${\overline{z}}$ as independent variables. Let
$f(z,{\overline{z}})=L_{\beta_{1}(z)\ldots\beta_{n}(z)}({\overline{z}})$ be a
hyperlogarithm in ${\overline{z}}$ with letters in
$\mathrm{FLT}\,_{\mathbb{F}}(z)$. For any $a\in{\mathbb{F}}$ we get
$\lim_{z\to
a}f(z,{\overline{z}})\in\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}({\overline{z}})$,
where the explicit result may depend on the direction in which $z$ approaches
$a$.

###### Proof.

We use induction over $n$. For $n=0$ we have $f=1$ and the claim is trivial.
If $n\geq 1$ then $f(z,0)=0$ and

$\partial_{\overline{z}}f(z,{\overline{z}})=\frac{1}{{\overline{z}}-\beta_{n}(z)}L_{\beta_{1}(z)\ldots\beta_{n-1}(z)}({\overline{z}}).$

By Brown’s algorithm, the hyperlogarithm on the right hand side can be
converted into a hyperlogarithm in $z$ (with ${\overline{z}}$-dependent
letters). By Lemma 6 it admits a log expansion at $z=a$. By induction the
constant term is in
$\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}({\overline{z}})$. The factor
$({\overline{z}}-\beta_{n}(z))^{-1}$ has an expansion at $z=a$ with constant
term $({\overline{z}}-\beta_{n}(a))^{-1}$ if $\beta_{n}(a)\neq\infty$. In this
case $\beta_{n}(a)\in{\mathbb{F}}$ and integration with respect to
${\overline{z}}$ gives
$f\in\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}({\overline{z}})$ plus
terms that vanish in the limit $z\to a$. If $\beta_{n}(a)=\infty$, the
constant term vanishes and $\lim_{z\to a}f(z,{\overline{z}})=0$. ∎

######  Example 17.

Consider $f(z,{\overline{z}})=L_{z-a}({\overline{z}})$ for $a\in{\mathbb{F}}$.
We get the regularized limit (see Section 3.2)

$\lim_{z\to a}f(z,{\overline{z}})=\lim_{z\to
a}\log(1-{\overline{z}}/(z-a))=\lim_{z\to
a}\log(-{\overline{z}}/(z-a))=\log({\overline{z}})+c=L_{0}({\overline{z}})+c$

for some constant $c\in{\mathbb{C}}$ whose value depends on the direction in
which $z$ approaches $a$.

######  Remark 18.

Note that due to possible singularities, taking limits in hyperlogarithms
cannot be performed by substitution (in general). As an example consider

$\displaystyle\lim_{z\to 0}L_{z0}({\overline{z}})=\lim_{z\to
0}L_{10}({\overline{z}}/z)=-\lim_{z\to
0}\mathrm{Li}\,_{2}({\overline{z}}/z)=\lim_{z\to
0}\frac{[\log(z/{\overline{z}})\pm\mathrm{i}\pi]^{2}}{2}-\zeta(2)=\frac{[\log({\overline{z}})\mp\mathrm{i}\pi]^{2}}{2}-\zeta(2)$
$\displaystyle\neq\;L_{00}({\overline{z}})=\frac{(\log{\overline{z}})^{2}}{2},$

where the sign ambiguity is related to monodromy. The subtle step in Brown’s
algorithm is to accurately take the limits $z\to 0$ that provide the
integration constants [16].

######  Proposition 19.

Let $f(z,{\overline{z}})=L_{\beta_{1}(z)\ldots\beta_{n}(z)}({\overline{z}})$
be a hyperlogarithm in ${\overline{z}}$ with letters in
$\mathrm{FLT}\,_{\mathbb{F}}(z)$. Then $f$ is a linear combination of
hyperlogarithms in $z$ with letters in
$\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$ and coefficients in
$\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}({\overline{z}})$.

###### Proof.

We consider $z$ and ${\overline{z}}$ as independent variables and use
induction over $n$ (following Brown’s algorithm). For $n=0$ the claim is
trivial.

By differentiation with respect to $z$ we get from I8 a sum of hyperlogarithms
in ${\overline{z}}$ of weight $n-1$ with factors
$\pm\partial_{z}\log(\beta_{k+1}-\beta_{k})$ for $k=0,\ldots,n$ (where
$\beta_{0}=0$ and $\beta_{n+1}={\overline{z}}$). By induction the
hyperlogarithms are linear combinations of hyperlogarithms in $z$ with letters
in $\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$ and coefficients in
$\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}({\overline{z}})$.

With Lemma 15 (1) we express $\beta_{k+1}-\beta_{k}$ for $k<n$ as product of
two FLTs over ${\mathbb{F}}$. Upon differentiation the log becomes a sum of
simple poles in $z$ at values in ${\mathbb{F}}$.

The term $k=n$ is only non-zero if $\beta_{n}$ is not constant. In this case
we get from Lemma 14 that the coefficient is a sum of simple poles with values
in $\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$.

By (11) both cases have primitives which are linear combinations of
hyperlogarithms in $z$ with letters in
$\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$. These hyperlogarithms vanish in
the limit $z\to 0$. To determine $f$ we need to add the integration constant
(in $z$) which is $\lim_{z\to 0}f(z,{\overline{z}})$. By Lemma 16 this limit
is in $\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}({\overline{z}})$ ∎

######  Example 20.

Consider $f(z,{\overline{z}})=L_{z-a}({\overline{z}})$, $a\in{\mathbb{F}}$,
from Example 17. We have

$f(z,{\overline{z}})=\log\Big{(}1-\frac{{\overline{z}}}{z-a}\Big{)}=\log\Big{[}\Big{(}1-\frac{z}{{\overline{z}}+a}\Big{)}\Big{(}1+\frac{{\overline{z}}}{a}\Big{)}\Big{(}1-\frac{z}{a}\Big{)}^{-1}\Big{]}=L_{{\overline{z}}+a}(z)+L_{-a}({\overline{z}})-L_{a}(z).$

Differentiation with respect to $z$ gives
$\partial_{z}f(z,{\overline{z}})=(z-{\overline{z}}-a)^{-1}-(z-a)^{-1}$. Upon
integration we get the first and the third term. The second term is
$\lim_{z\to 0}f(z,{\overline{z}})$.

To obtain an explicit result one can use the path convention at the beginning
of Section 3.4. Otherwise the result depends on the sheets where the
hyperlogarithms are evaluated. This will become insignificant when we pass to
single-valued functions.

###  4.3. Definition of generalized hyperlogarithms

Let ${\overline{z}}$ be the complex conjugate of $z$. Let

$\Sigma({\overline{z}})=\\{\beta_{1}({\overline{z}}),\beta_{2}({\overline{z}}),\ldots,\beta_{N}({\overline{z}})\\}\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$

be a finite set of FLTs over ${\mathbb{F}}$ in the variable ${\overline{z}}$
with subset of constants $\Sigma({\overline{z}})\cap{\mathbb{F}}$. We define
the ring of regular functions on
${\mathbb{C}}\backslash\Sigma({\overline{z}}):={\mathbb{C}}\backslash\\{z=\beta_{i}({\overline{z}}),i=1,\ldots,N\\}$
as the ring generated by $\mathcal{O}_{\Sigma({\overline{z}})}(z)$ and
$\mathcal{O}_{\overline{\Sigma({\overline{z}})\cap{\mathbb{F}}}}({\overline{z}})$,
see (9),

(47)
$\mathcal{O}_{\Sigma}(z,{\overline{z}})={\mathbb{C}}[z,(z-\beta({\overline{z}}))^{-1}_{\beta\in\Sigma({\overline{z}})},{\overline{z}},({\overline{z}}-{\overline{b}})^{-1}_{b\in\Sigma({\overline{z}})\cap{\mathbb{F}}}].$

We also define

(48)
$\mathcal{O}_{\mathbb{F}}(z)=\bigcup_{\Sigma\subset{\mathbb{F}}}\mathcal{O}_{\Sigma}(z)\quad\text{and}\quad\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})=\bigcup_{\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})}\mathcal{O}_{\Sigma}(z,{\overline{z}}).$

######  Example 21.

Reciprocals of bilinear forms in $z$ and ${\overline{z}}$ are in
$\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})$. For $a,b,c,d\in{\mathbb{F}}$,
$a\neq 0$,

$\frac{1}{az{\overline{z}}+bz+c{\overline{z}}+d}=\frac{1}{a}\cdot\frac{1}{{\overline{z}}+b/a}\cdot\frac{1}{z+(c{\overline{z}}+d)/(a{\overline{z}}+b)}\in\mathcal{O}_{\\{-\overline{b}/{\overline{a}},-(c{\overline{z}}+d)/(a{\overline{z}}+b)\\}}(z,{\overline{z}}).$

######  Remark 22.

By partial fraction decomposition first in $z$ and then (the coefficients) in
${\overline{z}}$ every $f\in\mathcal{O}_{\Sigma}(z,{\overline{z}})$ has a
unique representation as

$f(z,{\overline{z}})=\sum_{\beta({\overline{z}})\in\Sigma({\overline{z}}),b\in{\mathbb{F}},m,{\overline{m}}}c^{\beta,b}_{m,{\overline{m}}}(z-\beta({\overline{z}}))^{m}({\overline{z}}-\overline{b})^{\overline{m}},$

where the sum is finite with $m,{\overline{m}}\in{\mathbb{Z}}$ and
$m,{\overline{m}}<0$ if $\beta\neq 0,b\neq 0$, respectively. Because partial
fraction decomposition may generate (spurious) singularities in
${\overline{z}}$ we cannot assume that
$b\in\Sigma({\overline{z}})\cap{\mathbb{F}}$. By Lemma 15 (1), however, we
have $b\in{\mathbb{F}}$.

######  Definition 23.

Let ($\Sigma^{\ast}$ is the sets of words in $\Sigma$)

(49) $\mathcal{G}\mathcal{H}_{\Sigma}=\langle
L_{v}({\overline{z}})L_{w({\overline{z}})}(z),v\in\overline{\Sigma({\overline{z}})\cap{\mathbb{F}}}^{\ast},w({\overline{z}})\in\Sigma({\overline{z}})^{\ast}\rangle_{\mathcal{O}_{\Sigma}(z,{\overline{z}})}$

be the space of generalized hyperlogarithms on
${\mathbb{C}}\backslash\Sigma({\overline{z}})$. We use the notation
$\mathcal{G}\mathcal{H}^{\mathbb{C}}_{\Sigma}$ for the set of generalized
hyperlogarithms with coefficients in ${\mathbb{C}}$. The sets
$\mathcal{G}\mathcal{H}_{\mathbb{F}}$ and
$\mathcal{G}\mathcal{H}^{\mathbb{C}}_{\mathbb{F}}$ are the unions of
$\mathcal{G}\mathcal{H}_{\Sigma}$ and
$\mathcal{G}\mathcal{H}^{\mathbb{C}}_{\Sigma}$ over all finite
$\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$
(respectively).

By Theorem 24 (1) there exists a unique representation of any
$f\in\mathcal{G}\mathcal{H}_{\Sigma}$ as a linear combination of
hyperlogarithms $L_{v}({\overline{z}})L_{w({\overline{z}})}(z)$ with non-zero
canceled fractions as coefficients. The weight of $f$ is the maximum total
weight $|v|+|w({\overline{z}})|$ in this representation.

We suppress the ${\overline{z}}$ dependence of $\Sigma({\overline{z}})$ in
subscripts. In $\mathcal{O}_{\Sigma}(z,{\overline{z}})$ we denote both
arguments $z$ and ${\overline{z}}$ to avoid confusion with
$\mathcal{O}_{\Sigma}(z)$.

###  4.4. Properties of generalized hyperlogarithms

In analogy to Theorem 4 and Proposition 8 we obtain the following theorem.

######  Theorem 24.

Let $\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}$ be finite.

  1. (1)

$\mathcal{G}\mathcal{H}_{\Sigma}$ is a free
$\mathcal{O}_{\Sigma}(z,{\overline{z}})$-module which is closed under
multiplication. In particular, $\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}$
is a ${\mathbb{C}}$-algebra.

  2. (2)

The sequence

$0\longrightarrow\mathcal{H}\mathcal{L}_{\mathbb{F}}({\overline{z}})\longrightarrow\mathcal{G}\mathcal{H}_{\mathbb{F}}\stackrel{{\scriptstyle\partial_{z}}}{{\longrightarrow}}\mathcal{G}\mathcal{H}_{\mathbb{F}}\longrightarrow
0$

is exact. I.e. the kernel of $\partial_{z}$ in
$\mathcal{G}\mathcal{H}_{\mathbb{F}}$ is
$\mathcal{H}\mathcal{L}_{\mathbb{F}}({\overline{z}})$ and every
$f\in\mathcal{G}\mathcal{H}_{\mathbb{F}}$ has a primitive
$F\in\mathcal{G}\mathcal{H}_{\mathbb{F}}$ with $\partial_{z}F=f$.

  3. (3)

$\mathcal{G}\mathcal{H}_{\Sigma}$ is differentially simple. I.e. for every
$0\neq f\in\mathcal{G}\mathcal{H}_{\Sigma}$ there exists a differential
operator $D$ such that $Df=1$.

  4. (4)

$\mathcal{G}\mathcal{H}_{\mathbb{F}}$ and
$\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$ are stable under complex
conjugation,

(50)
$\overline{\mathcal{G}\mathcal{H}_{\mathbb{F}}}=\mathcal{G}\mathcal{H}_{\mathbb{F}},\quad\overline{\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}}=\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}.$

  5. (5)

$\mathcal{G}\mathcal{H}_{\mathbb{F}}$ and
$\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$ are stable under
transformations in $\mathrm{FLT}\,_{\mathbb{F}}$: For any
$f\in\mathcal{G}\mathcal{H}_{\mathbb{F}}$,
$g\in\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$, and
$\beta\in\mathrm{FLT}\,_{\mathbb{F}}$ we have

(51) $f(\beta(z))\in\mathcal{G}\mathcal{H}_{\mathbb{F}}\quad\text{and}\quad
g(\beta(z))\in\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}.$

###### Proof.

We consider $z$ and ${\overline{z}}$ as independent complex variables. For (1)
we apply Theorem 4 (1) to the hyperlogarithms in $z$. The coefficients are
hyperlogarithms in ${\overline{z}}$ on which we apply Theorem 4 (1) again.
This proves that $\mathcal{G}\mathcal{H}_{\Sigma}$ is free. With (15) it is
clear that $\mathcal{G}\mathcal{H}_{\Sigma}$ is closed under multiplication.

We fix ${\overline{z}}$. By Theorem 4 (2) the kernel of $\partial_{z}$ on
$\mathcal{G}\mathcal{H}_{\mathbb{F}}$ are functions which do not depend on
$z$. By statement (1) the space of these functions is
$\mathcal{H}\mathcal{L}_{\mathbb{F}}({\overline{z}})$. To prove the existence
of primitives in $\mathcal{G}\mathcal{H}_{\mathbb{F}}$ we use the explicit
inductive construction in the proof of Theorem 4 (2). To do so we need partial
fraction decomposition with respect to $z$ in
$\mathcal{O}_{\Sigma}(z,{\overline{z}})$ which can be achieved by repeatedly
using the formula

$\frac{1}{(z-\beta_{i}({\overline{z}}))(z-\beta_{j}({\overline{z}}))}=\frac{1}{(\beta_{i}({\overline{z}})-\beta_{j}({\overline{z}}))(z-\beta_{i}({\overline{z}}))}-\frac{1}{(\beta_{i}({\overline{z}})-\beta_{j}({\overline{z}}))(z-\beta_{j}({\overline{z}}))}.$

By Lemma 15 (1) it is clear that both terms on the right hand side are in
$\mathcal{O}_{\Sigma^{\prime}}(z,{\overline{z}})$ for some
$\Sigma^{\prime}({\overline{z}})\supseteq\Sigma({\overline{z}})$ which may
contain new (constant) singularities due to zeros of
$\beta_{i}({\overline{z}})-\beta_{j}({\overline{z}})$.

For (3) we follow the steps in the proof of Theorem 4 (3). Now, multiplication
with the common denominator $Q$ ensures that the maximum weight part of $Qf$
is a polynomial in $z$ and ${\overline{z}}$ with maximum degree
$(m,{\overline{m}})$ in $(z,{\overline{z}})$. We define
$D_{0}=\partial_{z}^{m}\partial_{\overline{z}}^{\overline{m}}Q$ and find that
$D_{0}f$ has weight $n$ but $\partial_{z}D_{0}f$ and
$\partial_{\overline{z}}D_{0}f$ have weight $\leq n-1$. If
$D_{0}f=c\in{\mathbb{C}}^{\times}$ then $D=D_{0}/c$. Otherwise
$\partial_{z}D_{0}f\neq 0$ or $\partial_{\overline{z}}D_{0}f\neq 0$. If
$\partial_{z}D_{0}f\neq 0$ (or $\partial_{\overline{z}}D_{0}f\neq 0$) then (by
induction) there exists a differential operator $D_{1}$ such that
$D=D_{1}\partial_{z}D_{0}$ (or $D=D_{1}\partial_{\overline{z}}D_{0}$) fulfills
$Df=1$.

For (4) we observe that (see Example 21) the complex conjugate of
$\mathcal{O}_{\Sigma}(z,{\overline{z}})$ can be written as
$\mathcal{O}_{\Sigma^{\prime}}(z,{\overline{z}})$ for some new set
$\Sigma^{\prime}({\overline{z}})$. So, it suffices to prove the second
identity in (50). The complex conjugate of
$L_{v}({\overline{z}})L_{w({\overline{z}})}(z)$ is
$L_{\overline{v}}(z)L_{\overline{w}(z)}({\overline{z}})$. The first factor is
in $\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$. By Proposition 19 we
can write the second factor in terms of hyperlogarithms in $z$ with letters in
$\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$ and coefficients in
$\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}({\overline{z}})$. Hence,
also the second factor is in
$\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$. By statement (1) the
product is in $\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$.

To show (5) we first observe that $\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})$
is stable under transformations in $\mathrm{FLT}\,_{\mathbb{F}}$. A generator
$1/(z-\beta_{i}({\overline{z}}))$ in
$\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})$ is transformed to
$1/(\beta(z)-\gamma({\overline{z}}))$ with
$\gamma=\beta_{i}\circ\overline{\beta}\in\mathrm{FLT}\,_{\mathbb{F}}$. The
result has a denominator which is bilinear in $z$ and ${\overline{z}}$. By
Example 21, it is in $\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})$.

By linearity it suffices to show that
$g(\beta(z))\in\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$ for
$g(z)=L_{v}({\overline{z}})L_{w({\overline{z}})}(z)$,
$v\in\overline{\Sigma({\overline{z}})\cap{\mathbb{F}}}^{\ast}$,
$w\in\Sigma^{\ast}$, see (49). We use Proposition 19 for
$\beta_{1}\ldots\beta_{n}\mapsto
w\circ\overline{\beta}\in\mathrm{FLT}\,_{\mathbb{F}}^{\ast}$ and
$(z,{\overline{z}})\mapsto({\overline{z}},\beta(z))$ to obtain that
$L_{w(\overline{\beta(z)})}(\beta(z))$ is a linear combination of
hyperlogarithms in ${\overline{z}}$ with letters in
$\mathrm{FLT}\,_{\mathbb{F}}(\beta(z))\subseteq\mathrm{FLT}\,_{\mathbb{F}}(z)$
and coefficients in
$\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}(\beta(z))$. By Brown’s
algorithm (see Section 4.2) the coefficiens are in
$\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}(z)$. Likewise
$L_{v}(\overline{\beta(z)})\in\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}({\overline{z}})$.
The result shuffles to a function in
$\overline{\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}}=\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$
(see statement (4)). ∎

######  Remark 25.

With statement (4) in the above theorem we see that the complex conjugate of
statement (2) holds for $\partial_{\overline{z}}$. In particular,
$\mathcal{G}\mathcal{H}_{\mathbb{F}}$ is closed under taking anti-primitives.

######  Example 26.

Every product of hyperlogarithms in $z$ and in ${\overline{z}}$ with letters
in ${\mathbb{F}}$ is in $\mathcal{G}\mathcal{H}_{\mathbb{F}}$,

$\mathcal{H}\mathcal{L}_{\mathbb{F}}(z)\mathcal{H}\mathcal{L}_{\mathbb{F}}({\overline{z}})\subset\mathcal{G}\mathcal{H}_{\mathbb{F}}.$

######  Example 27.

Extending Example 21 to generalized hyperlogarithms, we observe that every
iterated integral in forms with denominators which are bilinear in $z$ and
${\overline{z}}$ over ${\mathbb{F}}$ is in
$\mathcal{G}\mathcal{H}_{\mathbb{F}}$. This is clear from using Brown’s
algorithm as in the proof of Proposition 19.

In general, it is not easy to see if a given integral is in
$\mathcal{G}\mathcal{H}_{\mathbb{F}}$.

###  4.5. Singular decomposition

Any function $f\in\mathcal{G}\mathcal{H}_{\mathbb{F}}$ has a log-Laurent
expansion at $z=0$ whose coefficients have log-Laurent expansions at
${\overline{z}}=0$, see (24). In general, with every order in $z$ the pole
order in ${\overline{z}}$ increases. It is also possible to first expand $f$
in ${\overline{z}}$ and the coefficients in $z$. In this case one may get
increasingly negative orders in $z$. For GSVHs we are interested in functions
where both expansions are equal. This is equivalent to the existence of a
global limit for pole orders in $z$ and ${\overline{z}}$.

######  Definition 28.

A function $f$ admits a (simultaneous) log-Laurent expansion at
$z={\overline{z}}=0$ if

(52)
$f(z,{\overline{z}})=\sum_{\ell,{\overline{\ell}}=0}^{L}\sum_{m,{\overline{m}}=M}^{\infty}c_{\ell,{\overline{\ell}},m,{\overline{m}}}(\log
z)^{\ell}(\log{\overline{z}})^{\overline{\ell}}z^{m}{\overline{z}}^{\overline{m}},\quad
0\leq
L,M\in{\mathbb{Z}},\;c_{\ell,{\overline{\ell}},m,{\overline{m}}}\in{\mathbb{C}},$

in a neighborhood of $z={\overline{z}}=0$. The ${\mathbb{C}}$-algebra of
functions with log-Laurent expansions at $0$ is $\mathcal{R}^{0}$. If $L=M=0$
we say that $f$ is ${\mathbb{C}}$-analytic at 0, see Definition 35.

One may consider $\mathcal{R}^{0}$ as space of regular functions at 0. Note
that the definition of $\mathcal{R}^{0}$ does not depend on the sheets on
which the logarithms are evaluated. By (24) we get
$\mathcal{H}\mathcal{L}(z)\mathcal{H}\mathcal{L}({\overline{z}})\subset\mathcal{R}^{0}$.

######  Example 29.

The function
$\log(z-{\overline{z}})=L_{\overline{z}}(z)+L_{0}({\overline{z}})\pm\mathrm{i}\pi$
is not in $\mathcal{R}^{0}$.

In this section we derive a unique decomposition of a function
$f\in\mathcal{G}\mathcal{H}_{\Sigma}$ into a regular and a singular part such
that $f$ is regular at 0 if and only if its singular part vanishes. We need
the decomposition to prove Theorem 52 which, in turn, is essential to prove
the main theorem for GSVHs, Theorem 74.

######  Lemma 30.

Let $w({\overline{z}})$ be a word with letters in
$\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$ and
$\delta_{0}^{\mathrm{l}}w({\overline{z}})=0$. Then (the empty sum is zero)

(53)
$L_{0^{\\{L\\}}w({\overline{z}})}(z)=\sum_{\ell=0}^{L}\Big{(}g_{\ell}(z,{\overline{z}})\log^{\ell}(z)+\sum_{\genfrac{}{}{0.0pt}{}{w=u\beta
v}{\beta\neq 0=\beta(0)}}f_{\ell}^{\beta
v}({\overline{z}})L_{0^{\\{\ell\\}}\beta({\overline{z}})v({\overline{z}})}(z)\Big{)},$

where $u,v\in\mathrm{FLT}\,_{\mathbb{F}}^{\ast}$ and
$\beta\in\mathrm{FLT}\,_{\mathbb{F}}$. The functions
$g_{\ell}(z,{\overline{z}})\in\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$
are ${\mathbb{C}}$-analytic at $0$ with $g_{\ell}(0,{\overline{z}})=0$. The
functions $f_{\ell}^{\beta
v}({\overline{z}})\in\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}({\overline{z}})$
are anti-holomorphic at ${\overline{z}}=0$.

###### Proof.

The proof is by induction over the weight $n$ of $w({\overline{z}})$. The case
$n=0$ is trivial by (12).

Assume that $n\geq 1$ and
$w({\overline{z}})=x({\overline{z}})\beta_{n}({\overline{z}})$ with
$x\in\mathrm{FLT}\,_{\mathbb{F}}^{\ast}$ and
$\beta_{n}\in\mathrm{FLT}\,_{\mathbb{F}}$. By induction we obtain

(54)
$\partial_{z}L_{0^{\\{L\\}}x({\overline{z}})\beta_{n}({\overline{z}})}(z)=\frac{1}{z-\beta_{n}({\overline{z}})}\Big{(}\sum_{\ell=0}^{L}g_{\ell}(z,{\overline{z}})\log^{\ell}(z)+\sum_{\genfrac{}{}{0.0pt}{}{x=u\beta
v}{\beta\neq 0=\beta(0)}}f_{\ell}^{\beta
v}({\overline{z}})L_{0^{\\{\ell\\}}\beta({\overline{z}})v({\overline{z}})}(z)\Big{)}.$

If $\beta_{n}({\overline{z}})=0$ we use integration by parts $\ell+1$ times in
the first term of the sum over $\ell$, iteratively integrating
$g_{\ell}(z,{\overline{z}})/z$ (which is ${\mathbb{C}}$-analytic at $0$
because $g_{\ell}(0,{\overline{z}})=0$) and differentiating $\log^{\ell}(z)$.
With (3.5) we obtain a set of new functions
$\tilde{g}_{\ell}\in\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$ with
$\tilde{g}_{\ell}(0,{\overline{z}})=0$. Integration of the second term merely
adds a letter 0 to $v({\overline{z}})$. The integration constant is zero
because both sides of (53) vanish at $z=0$ (and any value of ${\overline{z}}$
in the neighborhood of 0). This gives the result for
$\beta_{n}({\overline{z}})=0$.

If $\beta_{n}(0)\neq 0$ then $1/(z-\beta_{n}({\overline{z}}))$ is
${\mathbb{C}}$-analytic at $0$ and the result follows like in the case
$\beta_{n}({\overline{z}})=0$.

We are left with the case $\beta_{n}({\overline{z}})\neq 0=\beta_{n}(0)$ which
implies $\beta_{n}({\overline{z}})=a{\overline{z}}/(b{\overline{z}}+1)$ for
some constants $a,b\in{\mathbb{F}}$, $a\neq 0$. In (54) we write

(55)
$g_{\ell}(z,{\overline{z}})=[g_{\ell}(z,{\overline{z}})-g_{\ell}(\beta_{n}({\overline{z}}),{\overline{z}})]+g_{\ell}(\beta_{n}({\overline{z}}),{\overline{z}}).$

Because $g_{\ell}(z,{\overline{z}})$ is ${\mathbb{C}}$-analytic,
$g_{\ell}(\beta_{n}({\overline{z}}),{\overline{z}})$ is anti-holomorphic at
${\overline{z}}=0$. By
$g_{\ell}\in\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$ we have

$g_{\ell}(\beta_{n}({\overline{z}}),{\overline{z}})=\sum_{v,w({\overline{z}})}c_{v,w({\overline{z}})}L_{v}({\overline{z}})L_{w({\overline{z}})}(\beta_{n}({\overline{z}})),\quad
c_{v,w({\overline{z}})}\in{\mathbb{C}}.$

With Brown’s algorithm and Lemma 15 (1) we obtain
$L_{w({\overline{z}})}(\beta_{n}({\overline{z}}))\in\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}({\overline{z}})$.
Therefore
$g_{\ell}(\beta_{n}({\overline{z}}),{\overline{z}})\in\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}({\overline{z}})$.
This implies
$g_{\ell}(z,{\overline{z}})-g_{\ell}(\beta_{n}({\overline{z}}),{\overline{z}})\in\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$.

By the theory of complex functions (or by explicit calculation) we see that

$g_{\ell}(z,{\overline{z}})-g_{\ell}(\beta_{n}({\overline{z}}),{\overline{z}})=[z-\beta_{n}({\overline{z}})]h_{\ell}(z,{\overline{z}})$

with functions $h_{\ell}$ which are ${\mathbb{C}}$-analytic at $0$. We follow
the argument of the case $\beta_{n}({\overline{z}})=0$ to see that integrating
the first term of (55) gives rise to a set of new functions
$\tilde{g}_{\ell}\in\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$ with
$\tilde{g}_{\ell}(0,{\overline{z}})=0$. Integrating the second term of (55)
gives
$\ell!g_{\ell}(\beta_{n}({\overline{z}}),{\overline{z}})L_{0^{\\{\ell\\}}\beta_{n}({\overline{z}})}(z)$
which is the term $v=e$ in the second sum of (53).

The terms $v\neq e$ are trivially obtained from integrating the second sum in
(54). ∎

Let $\mathcal{R}^{0}({\overline{z}})$ be the space of anti-analytic functions
in $\mathcal{R}^{0}$. Complementary to Definition 28 we define singular
functions at 0.

######  Definition 31.

For any finite set
$\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$ we
define the ${\mathbb{C}}$-algebra $\mathcal{S}^{0}_{\Sigma}$ of singular
functions at 0 as the $\mathcal{R}^{0}({\overline{z}})$-span of
hyperlogarithms $L_{0^{\\{\ell\\}}\beta({\overline{z}})v({\overline{z}})}(z)$
with $\beta({\overline{z}})\neq 0=\beta(0)$. So, every
$f\in\mathcal{S}^{0}_{\Sigma}$ has a unique representation as

(56) $f(z,{\overline{z}})=\sum_{\ell=0}^{L}\sum_{\genfrac{}{}{0.0pt}{}{\beta
v}{\beta\neq 0=\beta(0)}}f_{\ell}^{\beta
v}({\overline{z}})L_{0^{\\{\ell\\}}\beta({\overline{z}})v({\overline{z}})}(z),\quad\text{with
}L\geq 0,\;\beta\in\Sigma,\;v\in\Sigma^{\ast},\;f_{\ell}^{\beta
v}({\overline{z}})\in\mathcal{R}^{0}({\overline{z}}).$

We consider the $\mathcal{O}_{\Sigma}(z,{\overline{z}})$ modules
$\mathcal{O}_{\Sigma}(z,{\overline{z}})\mathcal{R}^{0}$ and
$\mathcal{O}_{\Sigma}(z,{\overline{z}})\mathcal{S}^{0}_{\Sigma}$. Note that a
function $f\in\mathcal{O}_{\Sigma}(z,{\overline{z}})\mathcal{R}^{0}$ does not
in general have an expansion (52). After the multiplication with a common
denominator $Q$, however, we get
$Qf\in{\mathbb{C}}[z,{\overline{z}}]\mathcal{R}^{0}=\mathcal{R}^{0}$.

The module $\mathcal{O}_{\Sigma}(z,{\overline{z}})\mathcal{R}^{0}$ is stable
under differentiation with respect to $z$ (because
$\partial_{z}\mathcal{O}_{\Sigma}(z,{\overline{z}})\subset\mathcal{O}_{\Sigma}(z,{\overline{z}})$
and $\partial_{z}\mathcal{R}^{0}\subset\mathcal{R}^{0}$) whereas
$\mathcal{O}_{\Sigma}(z,{\overline{z}})\mathcal{S}^{0}_{\Sigma}$ is not,
$\partial_{z}\mathcal{S}^{0}_{\Sigma}\ni\partial_{z}L_{0^{\\{\ell\\}}\beta({\overline{z}})}(z)=L_{0^{\\{\ell\\}}}(z)/(z-\beta({\overline{z}}))\notin\mathcal{O}_{\Sigma}(z,{\overline{z}})\mathcal{S}^{0}_{\Sigma}$.

######  Proposition 32.

For any finite set
$\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$ we
have

(57)
$\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})\mathcal{S}^{0}_{\Sigma}\cap\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})\mathcal{R}^{0}\cap\mathcal{G}\mathcal{H}_{\mathbb{F}}=\\{0\\}.$

###### Proof.

Consider
$f(z,{\overline{z}})=\sum_{w}f_{w}(z,{\overline{z}})L_{w({\overline{z}})}(z)\in\mathcal{X}:=\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})\mathcal{S}^{0}_{\Sigma}\cap\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})\mathcal{R}^{0}\cap\mathcal{G}\mathcal{H}_{\mathbb{F}}$,
where $w({\overline{z}})=0^{\\{\ell\\}}\beta({\overline{z}})v({\overline{z}})$
with $\beta({\overline{z}})\neq 0=\beta(0)$ (by
$f\in\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})\mathcal{S}^{0}_{\Sigma}$) and
$f_{w}\in\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}({\overline{z}})$
(by $f\in\mathcal{G}\mathcal{H}_{\mathbb{F}}$). We assume that $0\neq
f\in\mathcal{X}$ is a minimal counter-example with respect to ‘$\prec$’ in
Definition 2 for the weight $|w({\overline{z}})|$.

We consider $\partial_{z}f$ and $\partial_{\overline{z}}f$ which are in
$\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})\mathcal{R}^{0}\cap\mathcal{G}\mathcal{H}_{\mathbb{F}}$.
The differential operators act on $f_{w}(z,{\overline{z}})$ and on
$L_{w({\overline{z}})}(z)$. In the latter case we get from I8 a sum of
hyperlogarithms with one letter of $w({\overline{z}})$ removed times factors
in $\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})$. By Lemma 30 the
hyperlogarithms split into terms in
$\mathcal{R}^{0}\cap\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$ and
terms in
$\mathcal{S}^{0}_{\Sigma}\cap\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$.
We subtract all terms of the first type from $\partial_{z}f$ (or
$\partial_{\overline{z}}f$) to obtain an expression in
$\mathcal{R}^{0}\cap\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$ which
also is in $\mathcal{S}^{0}_{\Sigma}$. The expression is in $\mathcal{X}$ and
it has the coefficients $\partial_{z}f_{w}$ (or
$\partial_{\overline{z}}f_{w}$) in the highest weight part.

For any polynomial $0\neq P\in{\mathbb{C}}[z,{\overline{z}}]$ we get $0\neq
Pf\in\mathcal{X}$. From a given function in $\mathcal{X}$ we can hence
construct new functions in $\mathcal{X}$ where the coefficients of highest
weight are multiplied with polynomials and repeatedly differentiated with
$\partial_{z}$ and $\partial_{\overline{z}}$. From (the proof of) Theorem 24
(3) we construct a differential operator $D$ with $Df_{v}(z,{\overline{z}})=1$
for a fixed $v({\overline{z}})$ of maximum weight. We may hence assume
$f_{v}=1$ without restriction.

By induction, the differential operators $\partial_{z}$ and
$\partial_{\overline{z}}$ map $f$ to polynomials in $\log(z)$ with
coefficients in
$\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}({\overline{z}})$,
see (53). Integrating $\partial_{z}f$ provides hyperlogarithms with words
$0^{\\{\ell\\}}\beta({\overline{z}})$. Because
$f\in\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})\mathcal{S}^{0}_{\Sigma}$ we
get

$f(z,{\overline{z}})=\sum_{\ell=0}^{L}\sum_{\genfrac{}{}{0.0pt}{}{\beta({\overline{z}})\in\Sigma({\overline{z}})}{\beta\neq
0=\beta(0)}}f_{0^{\\{\ell\\}}\beta}(z,{\overline{z}})L_{0^{\\{\ell\\}}\beta({\overline{z}})}(z).$

Because $\partial_{z}f$ and $\partial_{\overline{z}}f$ are polynomials in
$\log(z)$ we get
$f_{0^{\\{L\\}}\beta}(z,{\overline{z}})=:f_{\beta}\in{\mathbb{C}}$. We assume
that $L$ is minimal, i.e. at least one $f_{\beta}\neq 0$.

We multiply $f\in\mathcal{O}_{\Sigma}(z,{\overline{z}})\mathcal{R}^{0}$ with
the common denominator $Q(z,{\overline{z}})=\sum_{k+{\overline{k}}\leq
d}c_{k,{\overline{k}}}z^{k}{\overline{z}}^{\overline{k}}$,
$c_{k,{\overline{k}}}\in{\mathbb{C}}$, so that $Qf\in\mathcal{R}^{0}$. Using
(17) we calculate the coefficient of $(\log z)^{L}$ in the log-Laurent
expansion of $Qf$,

$c_{L}(z,{\overline{z}})=\frac{Q(z,{\overline{z}})}{L!}\sum_{\genfrac{}{}{0.0pt}{}{\beta({\overline{z}})\in\Sigma({\overline{z}})}{\beta\neq
0=\beta(0)}}f_{\beta}L_{\beta({\overline{z}})}(z).$

The FLTs $\beta({\overline{z}})$ are of the form
$a{\overline{z}}/(b{\overline{z}}+1)$ with $a,b\in{\mathbb{F}}$, $a\neq 0$. We
expand $Q(z,{\overline{z}})$ and
$L_{\beta({\overline{z}})}(z)=\log(1-z(b{\overline{z}}+1)/a{\overline{z}})$
first in $z$ and then in ${\overline{z}}$ to obtain

$c_{L}(z,{\overline{z}})=-\frac{1}{L!}\sum_{\genfrac{}{}{0.0pt}{}{k,{\overline{k}}\geq
0}{k+{\overline{k}}\leq
d}}\sum_{k_{1}=1}^{\infty}\sum_{i=0}^{k_{1}}\sum_{\beta({\overline{z}})=\frac{a{\overline{z}}}{b{\overline{z}}+1}\in\Sigma({\overline{z}})}c_{k,{\overline{k}}}f_{\beta}z^{k+k_{1}}{\overline{z}}^{{\overline{k}}-k_{1}+i}\frac{\genfrac{(}{)}{0.0pt}{}{k_{1}}{i}b^{i}}{k_{1}a^{k_{1}}}.$

Large values of $k_{1}$ give poles in ${\overline{z}}$. We read off the
$z^{m}{\overline{z}}^{{\overline{m}}-m}$ coefficient of $c_{L}$,

(58)
$c_{L,m,{\overline{m}}-m}=-\frac{1}{L!}\sum_{\genfrac{}{}{0.0pt}{}{k,{\overline{k}}\geq
0}{k+{\overline{k}}\leq\min\\{d,{\overline{m}}\\}}}\sum_{\beta({\overline{z}})=\frac{a{\overline{z}}}{b{\overline{z}}+1}\in\Sigma({\overline{z}})}c_{k,{\overline{k}}}f_{\beta}\frac{\genfrac{(}{)}{0.0pt}{}{m-k}{{\overline{m}}-k-{\overline{k}}}b^{{\overline{m}}-k-{\overline{k}}}}{(m-k)a^{m-k}},\quad\text{if
}m>{\overline{m}}.$

Because $Qf\in\mathcal{R}^{0}$ we get $c_{L,m,{\overline{m}}-m}=0$ for fixed
${\overline{m}}$ and sufficiently large $m$. This leads to a system of linear
equations for $c_{k,{\overline{k}}}f_{\beta}$. With an infinite number of
values for $m$ we may consider $m$ as a variable (by interpolation). For
$k+{\overline{k}}={\overline{m}}$ the summand has poles $a^{k-m}/(m-k)$. For
all other values of $k,{\overline{k}}$ the summands are holomorphic in $m$. By
linear independence every pole term has to vanish separately. If
${\overline{m}}\leq d$,

$\sum_{\beta({\overline{z}})=\frac{a{\overline{z}}}{b{\overline{z}}+1}\in\Sigma({\overline{z}})}c_{k,{\overline{m}}-k}f_{\beta}\frac{1}{(m-k)a^{m-k}}\quad\text{for
all }k=0,\ldots,{\overline{m}}.$

Because at least one $c_{k,{\overline{m}}-k}\neq 0$ we get
$\sum_{\beta}f_{\beta}\,a^{k-m}=0$ which is a Vandermonde system in the
parameter $a$. The system has trivial kernel and we obtain

$\sum_{b:\beta({\overline{z}})=\frac{a{\overline{z}}}{b{\overline{z}}+1}\in\Sigma({\overline{z}})}f_{\beta}=0\quad\text{for
all }a.$

We conclude that there exists at least one
$\beta({\overline{z}})=\frac{a{\overline{z}}}{b{\overline{z}}+1}\in\Sigma({\overline{z}})$
with $b\neq 0$ and $f_{\beta}\neq 0$. Now, we also consider ${\overline{m}}$
as a variable. We multiply (58) with
$(m-{\overline{m}})!({\overline{m}}-d)!/(m-1)!$ and obtain for
${\overline{m}}>d$,

$0=\sum_{\genfrac{}{}{0.0pt}{}{k,{\overline{k}}\geq 0}{k+{\overline{k}}\leq
d}}\sum_{\genfrac{}{}{0.0pt}{}{\beta({\overline{z}})=\frac{a{\overline{z}}}{b{\overline{z}}+1}\in\Sigma({\overline{z}})}{b\neq
0}}\frac{c_{k,{\overline{k}}}f_{\beta}\,a^{k-m}b^{{\overline{m}}-k-{\overline{k}}}}{(m-k,k)(m-{\overline{m}}+1,{\overline{k}})({\overline{m}}-d+1,d-k-{\overline{k}})},$

where we used the Pochhammer symbol $(x,n)=(x+n-1)!/(x-1)!$. Note that terms
with $b=0$ vanish. We prove by induction over $n\leq d$ that
$c_{k,{\overline{k}}}=0$ for all $k+{\overline{k}}\leq n$. For $n=d$ this is a
contradiction to $Q\neq 0$.

For $n=0$ we consider the pole $1/{\overline{m}}$ which only exists in the
term $k={\overline{k}}=0$. By linear independence of pole terms we get
$\sum_{\beta:b\neq
0}c_{0,0}f_{\beta}a^{-m}b^{\overline{m}}/({\overline{m}}-d+1,d)=0$. After
multiplication with $({\overline{m}}-d+1,d)$ this is a Vandermonde system for
$a$ and $b$ in $m$ and ${\overline{m}}$, respectively. Because there exists a
$\beta$ with $b\neq 0$ and $f_{\beta}\neq 0$ we get $c_{0,0}=0$.

For general $n\leq d$ we need to consider poles in ${\overline{m}}$ and in
$m$. To do this we use a partial fraction basis in ${\overline{m}}$ whose
coefficients are $b^{{\overline{m}}-k-{\overline{k}}}$ times a function in $m$
(see Remark 22). Consider the pole $1/({\overline{m}}-n)$ which has
contributions from terms with $k+{\overline{k}}\leq n$. By induction we can
restrict ourselves to terms with $k+{\overline{k}}=n$ and obtain for the
coefficients

$\displaystyle 0$ $\displaystyle=\sum_{k=0}^{n}\sum_{\beta:b\neq
0}\frac{c_{k,n-k}f_{\beta}\,a^{k-m}b^{{\overline{m}}-n}}{(m-k,k)(m-n+1,n-k)(n-d+1,d-n)}$
$\displaystyle=\frac{(m-n+1,n-1)}{(n-d+1,d-n)}\sum_{k=0}^{n}\sum_{\beta:b\neq
0}\frac{c_{k,n-k}f_{\beta}\,a^{k-m}b^{{\overline{m}}-n}}{m-k}.$

The individual terms in the last sum vanish as they have different poles in
$m$. From the Vandermonde system we get $c_{k,n-k}f_{\beta}=0$ for all $\beta$
with $b\neq 0$ and all $k=0,\ldots,n$. The claim follows because at least one
$f_{\beta}\neq 0$. ∎

######  Remark 33.

The stronger
$\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})\mathcal{S}^{0}_{\Sigma}\cap\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})\mathcal{R}^{0}=\\{0\\}$
is also true. Here, we only need (57) which is easier to prove.

######  Theorem 34.

Let $\Sigma\subset\mathrm{FLT}\,_{\mathbb{F}}$ be finite and consider the
point $z=0$. Every $f\in\mathcal{G}\mathcal{H}_{\Sigma}$ has a unique
decomposition $f(z)=f_{\mathrm{r}}(z)+f_{\mathrm{s}}(z)$ into a regular part
$f_{\mathrm{r}}\in\mathcal{O}_{\Sigma}(z,{\overline{z}})\mathcal{R}^{0}\cap\mathcal{G}\mathcal{H}_{\mathbb{F}}$
and a singular part
$f_{\mathrm{s}}\in\mathcal{O}_{\Sigma}(z,{\overline{z}})\mathcal{S}^{0}_{\Sigma}\cap\mathcal{G}\mathcal{H}_{\mathbb{F}}$.
If $f\in\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}$ then
$f_{\mathrm{r}}\in\mathcal{R}^{0}\cap\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$
and
$f_{\mathrm{s}}\in\mathcal{S}^{0}_{\Sigma}\cap\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$.

###### Proof.

We first show the existence of the decomposition. By linearity it suffices to
show the existence for the individual terms
$f(z)=\phi(z)L_{v}({\overline{z}})L_{w({\overline{z}})}(z)$ in (49),
$\phi\in\mathcal{O}_{\Sigma}(z,{\overline{z}})$,
$v\in\overline{\Sigma({\overline{z}})\cap{\mathbb{F}}}^{\ast}$,
$w({\overline{z}})\in\Sigma({\overline{z}})^{\ast}$. If
$f\in\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}$ then $\phi\in{\mathbb{C}}$.
We use (53) for $L_{w({\overline{z}})}(z)$ and shuffle
$\phi(z)L_{v}({\overline{z}})$ with the result. Because
$L_{v}({\overline{z}})\in\mathcal{R}^{0}({\overline{z}})$, see (24), we find
that the terms with $\log^{\ell}(z)$ in (53) are in
$\mathcal{O}_{\Sigma}(z,{\overline{z}})\mathcal{R}^{0}\cap\mathcal{G}\mathcal{H}_{\mathbb{F}}$
and in $\mathcal{R}^{0}\cap\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$
if $f\in\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}$. Likewise we get that
the other terms are in
$\mathcal{O}_{\Sigma}(z,{\overline{z}})\mathcal{S}_{\Sigma}^{0}\cap\mathcal{G}\mathcal{H}_{\mathbb{F}}$
or in
$\mathcal{S}_{\Sigma}^{0}\cap\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$,
respectively.

For uniqueness we assume that
$f(z)=f_{\mathrm{s}}(z)+f_{\mathrm{r}}(z)=g_{\mathrm{s}}(z)+g_{\mathrm{r}}(z)$
are two decompositions. We get
$f_{\mathrm{s}}(z)-g_{\mathrm{s}}(z)=-f_{\mathrm{r}}(z)+g_{\mathrm{r}}(z)\in\mathcal{O}_{\Sigma}(z,{\overline{z}})\mathcal{S}^{0}_{\Sigma}\cap\mathcal{O}_{\Sigma}(z,{\overline{z}})\mathcal{R}^{0}\cap\mathcal{G}\mathcal{H}_{\mathbb{F}}$.
By Proposition 32 we get $f_{\mathrm{s}}(z)=g_{\mathrm{s}}(z)$ and
$f_{\mathrm{r}}(z)=g_{\mathrm{r}}(z)$. ∎

The above theorem implies that
$f\in\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}$ is in $\mathcal{R}^{0}$ if
and only if its singular part vanishes.

##  5\. Single-valued functions

In the previous section we defined functions which may have one-dimensional
singular loci in ${\mathbb{C}}$ (such as $\log(z-{\overline{z}})$). For
single-valuedness we now consider a finite set $\Sigma\subset{\mathbb{C}}$ of
point-like singularities.

In the context of pQFT, single-valuedness is a fundamental property of all
graphical functions ([17] with a proof in [14]). Graphical functions have
single-valued log-Laurent expansions at their singular points $0$, $1$, and
$\infty$ (conjectured in [14, 17] with a partial proof in [1]).

######  Definition 35.

Let $\Sigma$ be a finite set of points in ${\mathbb{C}}$. A function $f$ on
${\mathbb{C}}\backslash\Sigma$ has a single-valued log-Laurent expansion at
$a\in\Sigma$ if

(59)
$f(z)=\sum_{\ell=0}^{L_{a}}\sum_{m,{\overline{m}}=M_{a}}^{\infty}c_{\ell,m,{\overline{m}}}^{a}[\log(z-a)({\overline{z}}-{\overline{a}})]^{\ell}(z-a)^{m}({\overline{z}}-{\overline{a}})^{\overline{m}}$

in some neighborhood of $a$. A function $f$ has a single-valued log-Laurent
expansion at $\infty$ if

(60)
$f(z)=\sum_{\ell=0}^{L_{\infty}}\sum_{m,{\overline{m}}=-\infty}^{M_{\infty}}c_{\ell,m,{\overline{m}}}^{\infty}[\log
z{\overline{z}}]^{\ell}z^{m}{\overline{z}}^{\overline{m}}$

in some neighborhood of $\infty$. We say that $f$ is ${\mathbb{C}}$-analytic
(i.e. ${\mathbb{C}}$-valued real-analytic) at
$a\in{\mathbb{C}}\cup\\{\infty\\}$ if $L_{a}=M_{a}=0$.

The space of ${\mathbb{C}}$-analytic functions on
${\mathbb{C}}\backslash\Sigma$ with single-valued log-Laurent expansions at
$\Sigma\cup\\{\infty\\}$ is $\mathcal{S}\mathcal{V}_{\Sigma}$. We also define

$\mathcal{S}\mathcal{V}_{\mathbb{C}}=\bigcup_{\Sigma\subset{\mathbb{C}}}\mathcal{S}\mathcal{V}_{\Sigma}.$

######  Remark 36.

  1. (1)

If $f$ has a single-valued log-Laurent expansion at $a$ then $f$ has trivial
monodromy at $a$.

  2. (2)

Because single-valuedness is defined locally it is clear that
$\mathcal{S}\mathcal{V}_{\Sigma}$ is a bi-differential ${\mathbb{C}}$-algebra.

  3. (3)

In general, one expects that every $f\in\mathcal{S}\mathcal{V}_{\Sigma}$ has a
primitive in $\mathcal{S}\mathcal{V}_{\Sigma}$. Because integration has a non-
local character, it is not easy to construct these primitives in general.

  4. (4)

The space $\mathcal{S}\mathcal{V}_{\mathbb{C}}$ is invariant under complex
conjugation,
$\overline{\mathcal{S}\mathcal{V}_{\mathbb{C}}}=\mathcal{S}\mathcal{V}_{\mathbb{C}}$.

  5. (5)

The space $\mathcal{S}\mathcal{V}_{\mathbb{C}}$ is invariant under linear
transformations $z\mapsto az+b$ for $a,b\in{\mathbb{C}}$ and under the
inversion $z\mapsto 1/z$. These transformations generate
$\mathrm{FLT}\,_{\mathbb{C}}$; for any
$f\in\mathcal{S}\mathcal{V}_{\mathbb{C}}$ and
$\beta\in\mathrm{FLT}\,_{\mathbb{C}}$ we have
$f(\beta(z))\in\mathcal{S}\mathcal{V}_{\mathbb{C}}$.

######  Example 37.

From the expansion formulae (25), (26) for hyperlogarithms in
$\mathcal{H}\mathcal{L}_{\Sigma}$ we deduce that every single-valued linear
combination of products of hyperlogarithms in $z$ and ${\overline{z}}$ has
single-valued log-Laurent expansions at $\Sigma\cup\\{\infty\\}$. These
single-valued hyperlogarithms were constructed and studied by F. Brown in [4,
5]. Examples of low weights are ($D$ is the Bloch-Wigner dilogarithm (1))

$\log(z)+\log({\overline{z}})\in\mathcal{S}\mathcal{V}_{\\{0\\}},\quad\log(1-z)+\log(1-{\overline{z}})\in\mathcal{S}\mathcal{V}_{\\{1\\}},\quad
D(z)\in\mathcal{S}\mathcal{V}_{\\{0,1\\}}.$

######  Example 38.

The function
$f(z)=\log(z-{\overline{z}}+1)\in\mathcal{G}\mathcal{H}_{\\{{\overline{z}}-1\\}}$
is ${\mathbb{C}}$-analytic in ${\mathbb{C}}$. It fails to have a single-valued
log-Laurent expansion at infinity because
$f(z)=\log(z)+\log(1-({\overline{z}}-1)/z)$ has increasing powers of
${\overline{z}}$ in the expansion at $z=\infty$. Therefore
$f\notin\mathcal{S}\mathcal{V}_{\mathbb{C}}$.

######  Example 39.

The function $f(z)=\log(z{\overline{z}}+1)$ is in
$\mathcal{S}\mathcal{V}_{\emptyset}$. It is real-analytic in ${\mathbb{C}}$
and at infinity,

$\log(z{\overline{z}}+1)=\log(z{\overline{z}})+\sum_{m=-\infty}^{-1}\frac{(-1)^{m}}{m}(z{\overline{z}})^{m}\quad\text{for
}|z|>1.$

On the other hand, $\log(z{\overline{z}}-1)$ is singular on the unit circle
and hence not in $\mathcal{S}\mathcal{V}_{\mathbb{C}}$.

Generalizing the previous example (where
$\beta({\overline{z}})=-1/{\overline{z}}$) we define for any field
${\mathbb{F}}\subseteq{\mathbb{C}}$

(61)
$\mathrm{FLT}\,_{\mathbb{F}}^{\emptyset}({\overline{z}})=\Big{\\{}\beta({\overline{z}})=\frac{a{\overline{z}}+b}{c{\overline{z}}+d}\in\mathrm{FLT}\,_{\mathbb{F}},c\neq
0,\;\\{z=\beta({\overline{z}}),z\in{\mathbb{C}}\\}=\emptyset\Big{\\}}.$

Exclusion of the case $c=0$ avoids the situation of Example 38.

######  Example 40.

The function $f(z)=\log(az{\overline{z}}+bz+c{\overline{z}}+d)$ is in
$\mathcal{S}\mathcal{V}_{\emptyset}$ if
$-(c{\overline{z}}+d)/(a{\overline{z}}+b)\in\mathrm{FLT}\,_{\mathbb{C}}^{\emptyset}({\overline{z}})$.
By definition of $\mathrm{FLT}\,_{\mathbb{C}}^{\emptyset}({\overline{z}})$ it
is clear that $f$ is ${\mathbb{C}}$-analytic in ${\mathbb{C}}$. Because $a\neq
0$, $f$ has a single-valued log-Laurent expansion at infinity as in Example
39.

More interesting (and more important in pQFT, see Remark 80 (2)) are single-
valued functions with letters which are not in
$\mathrm{FLT}\,_{\mathbb{C}}^{\emptyset}({\overline{z}})$.

######  Example 41.

We will see in Example 71 that there exists a single-valued primitive of
$f(z)=D(z)/(z-{\overline{z}})$ (see also [11]). Note that the denominator of
$f$ vanishes on the real line. This singularity is lifted by the numerator, so
that $f\in\mathcal{S}\mathcal{V}_{\\{0,1\\}}$.

######  Example 42.

Graphical functions in even dimensions $\geq 4$ (conjecturally) are in
$\mathcal{S}\mathcal{V}_{\\{0,1\\}}$ [1].

######  Example 43.

The function $1/(z-2{\overline{z}})$ is ${\mathbb{C}}$-analytic in
${\mathbb{C}}\backslash\\{0\\}$. It has trivial monodromy at 0 but fails to
have a single-valued log-Laurant expansion at $0$. Therefore
$1/(z-2{\overline{z}})\notin\mathcal{S}\mathcal{V}_{\mathbb{C}}$

We define the projection onto the anti-residue-free part
$\pi_{\partial_{\overline{z}}}$ as the complex conjugate of (41),

$\pi_{\partial_{\overline{z}}}f(z)=f(z)-\sum_{a\in\Sigma}\frac{\overline{\mathrm{res}\,}_{a}f}{{\overline{z}}-{\overline{a}}},$

where $\overline{\mathrm{res}\,}_{a}f$ is the anti-residue of $f$ at $a$, i.e.
the coefficient $c^{a}_{0,0,-1}$ in (59) (whereas
$\mathrm{res}\,_{a}f=c^{a}_{0,-1,0}$).

In analogy to (39) we have on $\mathcal{S}\mathcal{V}_{\Sigma}$

(62)
$\mathrm{res}\,_{a}\partial_{z}=\overline{\mathrm{res}\,}_{a}\partial_{\overline{z}}$

for any $a\in\Sigma$ because both sides are the coefficient $c^{a}_{1,0,0}$ in
(59).

##  6\. Generalized single-valued hyperlogarithms

Generalized single-valued hyperlogarithms are generalized hyperlogarithms
which are single-valued in the sense of Definition 35.

###  6.1. Definitions and first results

######  Definition 44.

Let ${\mathbb{F}}=\overline{{\mathbb{F}}}$ be a quadratically closed number
field, see (45). Let
$\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$ be
finite, see (46). Then

(63)
$\mathcal{G}_{\Sigma}=\mathcal{G}\mathcal{H}_{\Sigma}\cap\mathcal{S}\mathcal{V}_{\mathbb{C}}$

is the space of generalized single-valued hyperlogarithms (GSVHs) on
${\mathbb{C}}\backslash(\Sigma({\overline{z}})\cap{\mathbb{F}})$, see Remark
46 (2). Likewise,
$\mathcal{G}_{\Sigma}^{\mathbb{C}}=\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}\cap\mathcal{S}\mathcal{V}_{\mathbb{C}}$
is the ${\mathbb{C}}$-algebra of GSVHs with constant coefficients. We also
define

$\mathcal{G}_{\mathbb{F}}=\bigcup_{\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})}\mathcal{G}_{\Sigma},\quad\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}=\bigcup_{\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})}\mathcal{G}_{\Sigma}^{\mathbb{C}}.$

The weight of $f\in\mathcal{G}_{\Sigma}$ is the weight of
$f\in\mathcal{G}\mathcal{H}_{\Sigma}$ in Definition 23.

######  Example 45.

Examples of GSVHs are in Examples 37, 39, 40, and 41. Graphical functions are
not GSVHs in general because they can fail to be generalized hyperlogarithms.

By Theorem 24 and Remark 36 it is clear that $\mathcal{G}_{\mathbb{F}}$ is a
bi-differential algebra which is closed under complex conjugation and
transformations in $\mathrm{FLT}\,_{\mathbb{F}}$.

######  Remark 46.

  1. (1)

In our context, constants have weight 0. One can set up a motivic version of
GSVHs where, e.g., $\zeta(3)$ has weight 3. In this context one may add the
motivic weight of constants to the weight of GSVHs which lifts the weight to a
grading on GSVHs. We will not pursue this here. The weight in the above
definition is a filtration.

  2. (2)

We will prove in Theorem 54 that single-valued functions in
$\mathcal{G}\mathcal{H}_{\Sigma}$ can only have singularities at
$\Sigma({\overline{z}})\cap{\mathbb{F}}$ (and at infinity). We have

$\mathcal{G}_{\Sigma}=\mathcal{G}\mathcal{H}_{\Sigma}\cap\mathcal{S}\mathcal{V}_{\Sigma\cap{\mathbb{F}}}.$

It is possible that a function in $\mathcal{G}_{\Sigma}$ with minimal set
$\Sigma({\overline{z}})$ is ${\mathbb{C}}$-analytic at a point in
$\Sigma({\overline{z}})\cap{\mathbb{F}}$, see Example 88.

We will prove that for any finite set $\Sigma_{0}\subset{\mathbb{C}}$ of
singularities the space
$\mathcal{G}_{\mathbb{F}}\cap\mathcal{S}\mathcal{V}_{\Sigma_{0}}$ is closed
under taking (anti-)primitives (Theorem 74). To do this we will use the
commutative hexagon in Figure 1 where we use the notation
$\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$,
$\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$,
$\partial_{z}\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ for
the set of functions obtained by differentiating
$\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$. The technical difficulty in the proof
is to show Corollary 60 which states that an expression in
$\partial_{z}\mathcal{G}_{\Sigma}$ with linearly independent coefficients has
single-valued individual terms. We need a series of auxiliary results.

######  Lemma 47.

Let $\beta\in\mathrm{FLT}\,_{\mathbb{F}}\backslash{\mathbb{F}}$ and
$a\in{\mathbb{F}}$ with $a=\beta({\overline{a}})$. Let $f(z,{\overline{z}})$
be ${\mathbb{C}}$-analytic at $a$. Then, there exists a function
$g(z,{\overline{z}})$ which is ${\mathbb{C}}$-analytic at $a$ such that

$\frac{f(z,{\overline{z}})}{z-\beta({\overline{z}})}=\frac{f(\beta({\overline{z}}),{\overline{z}})}{z-\beta({\overline{z}})}+g(z,{\overline{z}}).$

###### Proof.

Fix ${\overline{z}}$ in a neighborhood of ${\overline{a}}$ such that $f$ is
holomorphic at $z=\beta({\overline{z}})$ (we have $\beta({\overline{a}})=a$),
$f(z,{\overline{z}})=\sum_{k=0}^{\infty}f_{k}({\overline{z}})(z-\beta({\overline{z}}))^{k}$
with anti-holomorphic functions $f_{k}$. This gives

$g(z,{\overline{z}})=\sum_{k=0}^{\infty}f_{k+1}({\overline{z}})[z-a-(\beta({\overline{z}})-\beta({\overline{a}}))]^{k}.$

Because $\beta({\overline{a}})\neq\infty$ we get that
$\beta({\overline{z}})-\beta({\overline{a}})$ expands into a Taylor series in
${\overline{z}}-{\overline{a}}$ with low degree one. The above series expands
into a Taylor series at $(z,{\overline{z}})=(a,{\overline{a}})$. ∎

######  Example 48.

Consider $f(z)=\log(z{\overline{z}})/(z-1/{\overline{z}})$. Because
$\log(z{\overline{z}})\in\mathcal{G}_{\\{0\\}}$ we obtain from Lemma 47 that
$f$ is ${\mathbb{C}}$-analytic at $a$ for $|a|=1$ and hence at
${\mathbb{C}}\backslash\\{0\\}$. Because $(z-1/{\overline{z}})^{-1}$ is
${\mathbb{C}}$-analytic at $0$ and at $\infty$ we get
$f\in\mathcal{G}_{\\{0,{\overline{z}}^{-1}\\}}$.

######  Lemma 49.

Let $\beta_{i}\in\mathrm{FLT}\,_{\mathbb{F}}\backslash{\mathbb{F}}$ and
$a\in{\mathbb{F}}$ with $a=\beta_{i}({\overline{a}})$ for $i=1,\ldots,n$ and
$\beta_{i}\neq\beta_{j}$ for $i\neq j$. Let
$f_{i}\in\mathcal{G}_{\mathbb{F}}$. Then
$\sum_{i}f_{i}(z,{\overline{z}})/(z-\beta_{i}({\overline{z}}))$ has a single-
valued log-Laurent expansion (59) at $a$ if and only if
$f_{i}(\beta_{i}({\overline{z}}),{\overline{z}})=0$ for all $i$ in a
neighborhood of ${\overline{z}}={\overline{a}}$.

###### Proof.

By (59) we have

(64)
$f_{i}(z,{\overline{z}})=\sum_{\ell=0}^{L}\frac{[\log(z-a)({\overline{z}}-{\overline{a}})]^{\ell}}{[(z-a)({\overline{z}}-{\overline{a}})]^{-M}}f_{\ell,i}(z,{\overline{z}}),$

with ${\mathbb{C}}$-analytic functions $f_{\ell,i}$ at $a$ and
$L,M\in{\mathbb{Z}}$. Because $f_{i}\in\mathcal{G}_{\mathbb{F}}$ we can use
(49) to see that the coefficients of
$\log^{\ell}({\overline{z}}-{\overline{a}})$ in the expansion of $f_{i}$ are
in $\mathcal{G}\mathcal{H}_{\mathbb{F}}$. Hence
$f_{\ell,i}\in\mathcal{G}\mathcal{H}_{\mathbb{F}}$. Using Lemma 47 for
$f_{\ell,i}(z,{\overline{z}})$ gives

(65)
$\sum_{i=1}^{n}\frac{f_{i}(z,{\overline{z}})}{z-\beta_{i}({\overline{z}})}-g(z,{\overline{z}})=\sum_{\ell=0}^{L}\frac{[\log(z-a)({\overline{z}}-{\overline{a}})]^{\ell}}{[(z-a)({\overline{z}}-{\overline{a}})]^{-M}}\sum_{i=1}^{n}\frac{f_{\ell,i}(\beta_{i}({\overline{z}}),{\overline{z}})}{z-\beta_{i}({\overline{z}})}$

with a function $g$ which has a single-valued log-Laurent expansion at $a$.

Assume
$f_{1}(\beta_{1}({\overline{z}}),{\overline{z}})=\ldots=f_{n}(\beta_{n}({\overline{z}}),{\overline{z}})=0$
but there exists an $i$ such that
$f_{\ell,i}(\beta_{i}({\overline{z}}),{\overline{z}})\neq 0$ for some $\ell$.
Let $L^{\prime}$ be the maximum of these $\ell$. From (64) we get
($a=\beta_{i}({\overline{a}})$)

$0=\sum_{\ell=0}^{L^{\prime}}\frac{[\log(\beta_{i}({\overline{z}})-\beta_{i}({\overline{a}}))({\overline{z}}-{\overline{a}})]^{\ell}}{[(\beta_{i}({\overline{z}})-\beta_{i}({\overline{a}}))({\overline{z}}-{\overline{a}})]^{-M}}f_{\ell,i}(\beta_{i}({\overline{z}}),{\overline{z}}).$

Because the $f_{\ell,i}$ are ${\mathbb{C}}$-analytic at $a$ we get that the
$f_{\ell,i}(\beta_{i}({\overline{z}}),{\overline{z}})$ are anti-holomorphic at
${\overline{z}}={\overline{a}}$. The coefficient of
$\log^{L^{\prime}}({\overline{z}}-{\overline{a}})$ on the right hand side is
zero which implies
$f_{L^{\prime},i}(\beta_{i}({\overline{z}}),{\overline{z}})=0$. The
contradiction gives $f_{\ell,i}(\beta_{i}({\overline{z}}),{\overline{z}})=0$
for all $\ell,i$ and the right hand side of (65) vanishes.

If there exists an $i$ such that
$f_{i}(\beta({\overline{z}}),{\overline{z}})\neq 0$ then
$f_{\ell,i}(\beta_{i}({\overline{z}}),{\overline{z}})\neq 0$ for some $\ell$.
Let $L^{\prime}$ be the maximum of these $\ell$. To use Theorem 34 we shift
$(z,{\overline{z}})\mapsto(z+a,{\overline{z}}+{\overline{a}})$ in (65). By
partial fraction decomposition in $z$ we obtain a primitive $H$ of the right
hand side. From (12) we get

$H(z,{\overline{z}})=L^{\prime}![(\beta_{i}({\overline{z}}+{\overline{a}})-a){\overline{z}}]^{M}f_{L^{\prime},i}(\beta_{i}({\overline{z}}+{\overline{a}}),{\overline{z}}+{\overline{a}}))L_{0^{\\{L^{\prime}\\}}(\beta_{i}({\overline{z}}+{\overline{a}})-a)}(z)+\ldots.$

Because $f_{\ell,i}\in\mathcal{G}\mathcal{H}_{\mathbb{F}}$ we get
$H\in\mathcal{G}\mathcal{H}_{\mathbb{F}}$. The above term of $H$ is in
$\mathcal{S}^{0}_{\Sigma}$ for
$\Sigma({\overline{z}})={\\{0,\beta_{i}({\overline{z}}+{\overline{a}})-a\\}}$.

In Theorem 34 we get $H_{\mathrm{s}}\neq 0$. Hence
$H\notin\mathcal{R}^{0}\subset\mathcal{O}_{\Sigma}(z,{\overline{z}})\mathcal{R}^{0}$.
The expansion of $H$ in $z$ generates poles in ${\overline{z}}$ of increasing
order (see the proof of Proposition 32). This structure is preserved upon
differentiation (this is trivial for the maximum power of
$\log(z{\overline{z}})$)). Because
$g(z+a,{\overline{z}}+{\overline{a}})\in\mathcal{R}^{0}$ in (65) this implies
that
$\sum_{i}f_{i}(z+a,{\overline{z}}+{\overline{a}})/(z+a-\beta_{i}({\overline{z}}+{\overline{a}}))\notin\mathcal{R}^{0}$.
So, $\sum_{i}f_{i}(z,{\overline{z}})/(z-\beta_{i}({\overline{z}}))$ has no
single-valued log-Laurent expansion at $a$. ∎

######  Lemma 50.

Let $\beta_{i}({\overline{z}})=a_{i}{\overline{z}}+b_{i}$ with
$a_{i},b_{i}\in{\mathbb{F}}$, $a_{i}\neq 0$, for $i=1,\ldots,n$, and
$\beta_{i}\neq\beta_{j}$ for $i\neq j$. Let
$f_{i}\in\mathcal{G}_{\mathbb{F}}$. Then
$\sum_{i}f_{i}(z,{\overline{z}})/(z-\beta_{i}({\overline{z}}))$ has a single-
valued log-Laurent expansion (60) at $\infty$ if and only if
$f_{i}(\beta_{i}({\overline{z}}),{\overline{z}})=0$ for all $i$ in a
neighborhood of ${\overline{z}}=\infty$.

###### Proof.

Let
$g(z,{\overline{z}})=\sum_{i}f_{i}(1/z,1/{\overline{z}})/(1/z-\beta_{i}(1/{\overline{z}}))$.
We need to study $g$ at $z={\overline{z}}=0$. We get

$g(z,{\overline{z}})=\sum_{i=1}^{n}\frac{f_{i}(1/z,1/{\overline{z}})}{1/z-a_{i}/{\overline{z}}-b_{i}}=\sum_{i=1}^{n}\frac{-z{\overline{z}}f_{i}(1/z,1/{\overline{z}})}{(b_{i}{\overline{z}}+a_{i})(z-{\overline{z}}/(b_{i}{\overline{z}}+a_{i}))}.$

Consider
$\tilde{f}_{i}=-z{\overline{z}}f_{i}(1/z,1/{\overline{z}})/(b_{i}{\overline{z}}+a_{i})$
and
$\tilde{\beta}_{i}({\overline{z}})={\overline{z}}/(b_{i}{\overline{z}}+a_{i})$.
We have $\tilde{f}_{i}\in\mathcal{G}_{\mathbb{F}}$ by Theorem 24 (5) and
$\tilde{\beta}_{i}(0)=0$ because $a_{i}\neq 0$. With Lemma 49 we obtain that
$g$ has a single-valued log-Laurent expansion at 0 if and only if

$-\frac{\tilde{\beta}_{i}({\overline{z}}){\overline{z}}f_{i}(1/\tilde{\beta}_{i}({\overline{z}}),1/{\overline{z}})}{b_{i}{\overline{z}}+a_{i}}=-\frac{{\overline{z}}^{2}f_{i}(a_{i}/{\overline{z}}+b_{i},1/{\overline{z}})}{(b_{i}{\overline{z}}+a_{i})^{2}}=0$

for all $i$ in a neighborhood of ${\overline{z}}=0$. This is equivalent to
$f_{i}(a_{i}{\overline{z}}+b_{i},{\overline{z}})=0$ for all $i$ in a
neighborhood of ${\overline{z}}=\infty$. ∎

######  Example 51.

Consider $f(z)=D(z)/(z-{\overline{z}})$, see Example 41. By Example 37 we have
$D\in\mathcal{G}_{\\{0,1\\}}$. If $D=D(z,{\overline{z}})$ is considered as a
function in $z,{\overline{z}}$ then $D({\overline{z}},{\overline{z}})=0$ in
the neighborhood of any
${\overline{z}}={\overline{a}}\in{\mathbb{R}}\cup\\{\infty\\}$. From Lemma 47
we get that $f$ is ${\mathbb{C}}$-analytic at
${\mathbb{R}}\backslash\\{0,1\\}$ and hence in
${\mathbb{C}}\backslash\\{0,1\\}$. By Lemmas 49 and 50 the function $f$ has
single-valued log-Laurent expansions at $z=0,1,\infty$. So,
$f\in\mathcal{G}_{\\{0,1,{\overline{z}}\\}}$.

######  Theorem 52.

Let $\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$
be finite. Let $\mathcal{B}_{\Sigma}$ be a ${\mathbb{C}}$-basis of
$\mathcal{O}_{\Sigma}(z,{\overline{z}})$, see (47). Every
$f\in\mathcal{G}_{\Sigma}$ has a unique $\mathcal{B}_{\Sigma}$ representation
as

(66)
$f(z)=\sum_{\phi\in\mathcal{B}_{\Sigma}}\phi(z)h_{\phi}(z)\quad\text{with}\quad
h_{\phi}\in\mathcal{G}_{\Sigma}^{\mathbb{C}}.$

###### Proof.

Existence and uniqueness of a $\mathcal{B}_{\Sigma}$ representation with
$h_{\phi}\in\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}$ follow from Theorem
24 (1). We need to show that $h_{\phi}\in\mathcal{G}_{\Sigma}^{\mathbb{C}}$.

We first show that $h_{\phi}\in\mathcal{R}^{0}$. From
$f\in\mathcal{G}_{\Sigma}$ and Theorem 34 we get that in the decomposition of
$f$ we have $0=f_{\mathrm{s}}=\sum_{\phi}\phi h_{\phi,\mathrm{s}}$ with
$h_{\phi}=h_{\phi,\mathrm{r}}+h_{\phi,\mathrm{s}}$ and
$h_{\phi,\mathrm{r}},h_{\phi,\mathrm{s}}\in\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$.
From Theorem 24 (1) we get $h_{\phi,\mathrm{s}}=0$ by linear independence of
the $\phi(z)$. Therefore $h_{\phi}=h_{\phi,\mathrm{r}}\in\mathcal{R}^{0}$.

Consider the monodromy $\mathcal{M}_{0}$ at 0, see Section 3.4. Because
$f\in\mathcal{S}\mathcal{V}_{\mathbb{C}}$ we have
$f=\mathcal{M}_{0}f=\sum_{\phi}\phi\mathcal{M}_{0}h_{\phi}$ with
$\mathcal{M}_{0}h_{\phi}\in\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}$. By
linear independence of the $\phi$ we get $\mathcal{M}_{0}h_{\phi}=h_{\phi}$.
We rearrange the (multivalued) log-Laurent expansion of $h_{\phi}$ at $z=0$ so
that it has the terms $(\log
z{\overline{z}})^{\ell}(\log{\overline{z}})^{\overline{\ell}}z^{m}{\overline{z}}^{\overline{m}}$.
We get $\mathcal{M}_{0}(\log
z{\overline{z}})^{\ell}(\log{\overline{z}})^{\overline{\ell}}=(\log
z{\overline{z}})^{\ell}(\log{\overline{z}})^{\overline{\ell}}-2\pi\mathrm{i}{\overline{\ell}}(\log{\overline{z}})^{{\overline{\ell}}-1}$
plus terms with lower powers in $\log{\overline{z}}$. If there exist terms
with ${\overline{\ell}}\geq 1$ we consider the two leading powers in
$\log{\overline{z}}$ yielding ${\overline{\ell}}=0$. By contradiction,
$h_{\phi}$ has a single-valued log-Laurent expansion at 0.

For general $a\in{\mathbb{C}}$ we use $f(z+a)\in\mathcal{G}_{\mathbb{F}}$
which follows from Theorem 24 (5) and Remark 36 (5). We get that
$h_{\phi}(z+a)$ has a single-valued log-Laurent expansion at 0. Equivalently,
$h_{\phi}(z)$ has a single-valued log-Laurent expansion at $a$. Considering
$f(1/z)$ we find that $h_{\phi}(1/z)$ has a single-valued log-Laurent
expansion at 0. Therefore $h_{\phi}(z)$ has a single-valued log-Laurent
expansion at $\infty$.

In the neighborhood of the expansion points single-valued log-Laurent
expansions are ${\mathbb{C}}$-analytic. By compactness of
${\mathbb{C}}\cup\\{\infty\\}$ it follows that $h_{\phi}$ only has a finite
number of singularities. Hence
$h_{\phi}\in\mathcal{S}\mathcal{V}_{\mathbb{C}}$. This implies
$h_{\phi}\in\mathcal{G}_{\Sigma}^{\mathbb{C}}=\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}\cap\mathcal{S}\mathcal{V}_{\mathbb{C}}$.
∎

######  Remark 53.

It is possible to define
$\delta_{\beta}^{\mathrm{dR}}=\mathrm{res}\,_{\beta}\partial_{z}$ on
$\mathcal{G}_{\Sigma}^{\mathbb{C}}$ for any
$\beta({\overline{z}})\in\Sigma({\overline{z}})$ in analogy to (19). Theorem
52 implies that $\delta_{\beta}^{\mathrm{dR}}$ is an endomorphism of
$\mathcal{G}_{\Sigma}^{\mathbb{C}}$. To prove this property is a core
difficulty in this article. In Corollary 60 we prove the stronger result that
$(z-\beta({\overline{z}}))^{-1}\delta_{\beta}^{\mathrm{dR}}$ maps
$\mathcal{G}_{\Sigma}^{\mathbb{C}}$ into $\mathcal{G}_{\Sigma}$.

######  Theorem 54.

let $\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$
be finite. Then
$\mathcal{G}_{\Sigma}=\mathcal{G}\mathcal{H}_{\Sigma}\cap\mathcal{S}\mathcal{V}_{\Sigma\cap{\mathbb{F}}}$
and
$\mathcal{G}_{\Sigma}^{\mathbb{C}}=\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}\cap\mathcal{S}\mathcal{V}_{\Sigma\cap{\mathbb{F}}}$.
If $f\in\mathcal{G}_{\Sigma}^{\mathbb{C}}$ then $M_{a}=0$ in the single-valued
log-Laurent expansions (59), (60) at
$a\in(\Sigma({\overline{z}})\cap{\mathbb{F}})\cup\\{\infty\\}$.

###### Proof.

Any function in $\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ has single-valued log-
Laurent expansions with $M_{a}=0$. For the expansion in $z$ this follows from
(49), Lemma 6, and (26). By complex conjugation, Theorem 24 (4), this also
holds for the expansion in ${\overline{z}}$. So, we only need to prove the
first statement.

We first show that
$\mathcal{G}_{\Sigma}^{\mathbb{C}}=\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}\cap\mathcal{S}\mathcal{V}_{\Sigma\cap{\mathbb{F}}}$.
We trivially have
$\mathcal{G}_{\Sigma}^{\mathbb{C}}\supseteq\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}\cap\mathcal{S}\mathcal{V}_{\Sigma\cap{\mathbb{F}}}$.
For
$\mathcal{G}_{\Sigma}^{\mathbb{C}}\subseteq\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}\cap\mathcal{S}\mathcal{V}_{\Sigma\cap{\mathbb{F}}}$
we fix $a\in{\mathbb{C}}\backslash(\Sigma({\overline{z}})\cap{\mathbb{F}})$.
Let $f\in\mathcal{G}_{\Sigma}^{\mathbb{C}}$ have weight $n$. We prove by
induction over $n$ that $f$ is ${\mathbb{C}}$-analytic at $a$. For $n=0$ the
function $f$ is constant and the claim is trivial. If $n\geq 1$ then
$\partial_{z}f(z)=g(z)/(z-\beta({\overline{z}}))\in\mathcal{G}_{\Sigma}$ with
$\beta({\overline{z}})\in\Sigma({\overline{z}})$ and
$g\in\mathcal{G}_{\Sigma}^{\mathbb{C}}$ with weight $n-1$. By induction $g$ is
${\mathbb{C}}$-analytic at $a$.

For $\beta$ we have two possibilities: If $a\neq\beta({\overline{a}})$ then
$(z-\beta({\overline{z}}))^{-1}$ is ${\mathbb{C}}$-analytic at $a$. Therefore
$\partial_{z}f$ is ${\mathbb{C}}$-analytic at $a$. Otherwise
$a=\beta({\overline{a}})\notin\Sigma({\overline{z}})\cap{\mathbb{F}}$.
Therefore $\beta\notin{\mathbb{F}}$. We use Lemma 49 yielding
$g(\beta({\overline{z}}),{\overline{z}})=0$ in a neighborhood of
${\overline{z}}={\overline{a}}$. From Lemma 47 we get that $\partial_{z}f$ is
${\mathbb{C}}$-analytic at $a$.

Consider the single-valued log-Laurent expansion of
$f\in\mathcal{G}_{\Sigma}^{\mathbb{C}}$ at $a$. By explicit differentiation
with respect to $z$ we see that a ${\mathbb{C}}$-analytic $\partial_{z}f$
comes from a ${\mathbb{C}}$-analytic $f$ up to a pole term in
${\mathbb{C}}[1/({\overline{z}}-{\overline{a}})]$. The pole term is zero
because $M_{a}=0$. Hence
$f\in\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}\cap\mathcal{S}\mathcal{V}_{\Sigma\cap{\mathbb{F}}}$.

Now, let $f\in\mathcal{G}_{\Sigma}$ and fix
$a\notin\Sigma({\overline{z}})\cap{\mathbb{F}}$. We need to show that $f$ is
${\mathbb{C}}$-analytic at $a$. Consider a basis of
$\mathcal{O}_{\Sigma}(z,{\overline{z}})$ with each element in
$\mathcal{O}_{\Sigma}(z,{\overline{z}})$. From Theorem 52 we get
$h_{\phi}\in\mathcal{G}_{\Sigma}^{\mathbb{C}}$. Hence,
$h_{\phi}\in\mathcal{S}\mathcal{V}_{\Sigma\cap{\mathbb{F}}}$. Therefore
$h_{\phi}$ is ${\mathbb{C}}$-analytic at $a$.

We consider $z$ and ${\overline{z}}$ as independent variables. The function
$\phi\in\mathcal{O}_{\Sigma}(z,{\overline{z}})$ has poles at
$\\{z=\beta({\overline{z}}),\beta({\overline{z}})\in\Sigma({\overline{z}})\\}$
and at
$\\{{\overline{z}}={\overline{b}},b\in\Sigma({\overline{z}})\cap{\mathbb{F}}\\}$.
Because $a\neq\beta({\overline{z}})$, $\phi$ is holomorphic at $z=a$. Hence
$f$ is holomorphic at $a$. Likewise, $\phi$ is anti-holomorphic at
${\overline{z}}={\overline{a}}$ because $z\neq\beta({\overline{a}})$ and
${\overline{a}}\neq{\overline{b}}$ for all
$b\in\Sigma({\overline{z}})\cap{\mathbb{F}}$. So, $f$ is anti-holomorphic at
${\overline{z}}={\overline{a}}$. Because $f$ has a single-valued log-Laurent
expansion at $a$ we get that $f$ is ${\mathbb{C}}$-analytic at $a$. ∎

######  Definition 55.

Let $\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$
be finite. With $\mathrm{FLT}\,_{\mathbb{F}}^{\emptyset}({\overline{z}})$ as
in (61) we define

(67)
$\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})={\mathbb{C}}[z,{\overline{z}},((z-\beta({\overline{z}}))^{-1})_{\beta({\overline{z}})\in\Sigma({\overline{z}})\cap\mathrm{FLT}\,_{\mathbb{F}}^{\emptyset}({\overline{z}})},((z-b)^{-1},({\overline{z}}-{\overline{b}})^{-1})_{b\in\Sigma({\overline{z}})\cap{\mathbb{F}}}].$

######  Lemma 56.

The weight zero piece of $\mathcal{G}_{\Sigma}$ is
$\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})$:
$\mathcal{G}_{\Sigma}\cap\mathcal{O}_{\Sigma}(z,{\overline{z}})=\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})$.

###### Proof.

We have
$\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})\subseteq\mathcal{G}_{\Sigma}\cap\mathcal{O}_{\Sigma}(z,{\overline{z}})$
in analogy to Example 40.

Assume that a canceled fraction
$f\in\mathcal{G}_{\Sigma}\cap\mathcal{O}_{\Sigma}(z,{\overline{z}})$ has a
factor $z-\beta({\overline{z}})$ with
$\beta({\overline{z}})\in\Sigma({\overline{z}})\backslash(\mathrm{FLT}\,_{\mathbb{F}}^{\emptyset}({\overline{z}})\cup{\mathbb{F}})$
in the denominator,

$f(z,{\overline{z}})=\frac{P(z,{\overline{z}})}{(z-\beta({\overline{z}}))Q(z,{\overline{z}})},$

with polynomials $P,Q\in{\mathbb{C}}[z,{\overline{z}}]$. We have
$P(z,{\overline{z}})/(z-\beta({\overline{z}}))\in\mathcal{G}_{\Sigma}$. By
definition of $\mathrm{FLT}\,_{\mathbb{F}}^{\emptyset}$ there either exists an
$a\in{\mathbb{C}}$ with $a=\beta({\overline{a}})$ or $\beta$ is linear but not
constant. In any case we get from Lemmas 49 and 50 that
$P(\beta({\overline{z}}),{\overline{z}})=0$ in some neighborhood of
${\overline{z}}={\overline{a}}$ or $\infty$ (respectively). Therefore, the
factor $z-\beta({\overline{z}})$ cancels in $f$. By contradiction we obtain
$f\in\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})$. ∎

######  Definition 57.

Let $\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$
be finite. Then

(68)
$\partial_{z}\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}=\langle(z-\beta({\overline{z}}))^{-1},\,\beta({\overline{z}})\in\Sigma({\overline{z}})\rangle_{\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}}$

is the space of generalized hyperlogarithms with simple pole coefficients. In
analogy to (63) we define
$\partial_{z}\mathcal{G}_{\Sigma}^{\mathbb{C}}=\partial_{z}\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}\cap\mathcal{S}\mathcal{V}_{\mathbb{C}}$.
Moreover,
$\partial_{\overline{z}}\partial_{z}\mathcal{G}_{\Sigma}^{\mathbb{C}}=\partial_{\overline{z}}(\partial_{z}\mathcal{G}_{\Sigma}^{\mathbb{C}})$
and

$\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}=\bigcup_{\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})}\partial_{z}\mathcal{G}_{\Sigma}^{\mathbb{C}},\quad\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}=\overline{\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}},\quad\text{and}\quad\partial_{\overline{z}}\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}=\bigcup_{\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})}\partial_{\overline{z}}\partial_{z}\mathcal{G}_{\Sigma}^{\mathbb{C}}.$

Clearly,
$\partial_{z}(\mathcal{G}_{\mathbb{F}}^{\mathbb{C}})\subseteq\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$.
We also have
$\partial_{\overline{z}}(\mathcal{G}_{\mathbb{F}}^{\mathbb{C}})\subseteq\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$,
see I8 in Section 2 and Lemmas 14 and 15. We will show that equality holds
(Corollary 64) which also implies that
$\partial_{\overline{z}}\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}=\partial_{z}(\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}})$.

######  Proposition 58.

Let $\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$
be finite and let $\mathcal{B}_{\Sigma}$ be a ${\mathbb{C}}$-basis of
$\langle(z-\beta({\overline{z}}))^{-1},\beta({\overline{z}})\in\Sigma({\overline{z}})\rangle_{\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})}$.
If
$f\in\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})\partial_{z}\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}\cap\mathcal{S}\mathcal{V}_{\mathbb{C}}\subseteq\mathcal{G}_{\Sigma}$,

$f(z)=\sum_{\phi\in\mathcal{B}_{\Sigma}}\phi(z)h_{\phi}(z),\quad
h_{\phi}\in\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}},$

then $\phi h_{\phi}\in\mathcal{S}\mathcal{V}_{\mathbb{C}}$.

###### Proof.

The statement of the theorem is stable under basis transformations. Therefore,
we can assume without restriction that $\mathcal{B}_{\Sigma}$ is the basis of
partial fractions in $z$ and ${\overline{z}}$ defined in Remark 22.

Let
$f(z)=\sum_{\beta,b,m,{\overline{m}}}g^{\beta,b}_{m,{\overline{m}}}(z,{\overline{z}})(z-\beta({\overline{z}}))^{m}({\overline{z}}-{\overline{b}})^{\overline{m}}\in\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})\partial_{z}\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}\cap\mathcal{S}\mathcal{V}_{\mathbb{C}}$
where the sum is finite with
$g^{\beta,b}_{m,{\overline{m}}}\in\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}$,
$\beta({\overline{z}})\in\Sigma({\overline{z}})$, $b\in{\mathbb{F}}$,
$m,{\overline{m}}\in{\mathbb{Z}}$. From Theorem 52 we get
$g^{\beta,b}_{m,{\overline{m}}}\in\mathcal{G}_{\Sigma}^{\mathbb{C}}$.

Fix $a\in{\mathbb{C}}$. If $a\neq\beta\in{\mathbb{F}}$ or
$\beta({\overline{a}})\neq a$ then $(z-\beta({\overline{z}}))^{-1}$ is
${\mathbb{C}}$-analytic at $a$. In this case
$g^{\beta,b}_{m,{\overline{m}}}(z,{\overline{z}})(z-\beta({\overline{z}}))^{m}({\overline{z}}-{\overline{b}})^{\overline{m}}$
inherits the single-valued log-Laurent expansion from
$g^{\beta,b}_{m,{\overline{m}}}$. This also holds in the case $\beta=a$.

By subtracting these terms from the sum we obtain a reduced sum
$\sum_{\beta,b,{\overline{m}}}g^{\beta,b}_{-1,{\overline{m}}}(z,{\overline{z}})(z-\beta({\overline{z}}))^{-1}({\overline{z}}-{\overline{b}})^{\overline{m}}$
where all $\beta$ have the property $\beta\notin{\mathbb{F}}$ and
$\beta({\overline{a}})=a$. We have $m=-1$ because
$(z-\beta({\overline{z}}))^{-1}\notin\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})$.
The reduced sum has a single-valued log-Laurent expansion at $a$ (because $f$
and all subtracted terms have such an expansion). Lemma 49 for $n=1$ and
$f_{1}(z,{\overline{z}})=\sum_{b,{\overline{m}}}g^{\beta,b}_{-1,{\overline{m}}}(z,{\overline{z}})({\overline{z}}-{\overline{b}})^{\overline{m}}$
gives $f_{1}(\beta({\overline{z}}),{\overline{z}})=0$ in some neighborhood of
${\overline{z}}={\overline{a}}$. We have
$g^{\beta,b}_{-1,{\overline{m}}}(\beta({\overline{z}}),{\overline{z}})\in\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}({\overline{z}})$
and by linear independence (Theorem 4 (1)) we get
$g^{\beta,b}_{-1,{\overline{m}}}(\beta({\overline{z}}),{\overline{z}})=0$ for
all $b$, ${\overline{m}}$. Lemma 49 for $n=1$ and
$f_{1}(z,{\overline{z}})=g^{\beta,b}_{-1,{\overline{m}}}(z,{\overline{z}})({\overline{z}}-{\overline{b}})^{\overline{m}}$
gives that
$g^{\beta,b}_{-1,{\overline{m}}}(z,{\overline{z}})(z-\beta({\overline{z}}))^{-1}({\overline{z}}-{\overline{b}})^{\overline{m}}$
has a single-valued log-Laurent expansion at $a$.

At infinity we observe that terms with $\beta\in{\mathbb{F}}$ or
$\beta(\infty)\neq\infty$ trivially have single-valued log-Laurent expansions
at $\infty$. We reduce to a sum
$\sum_{\beta,b,{\overline{m}}}g^{\beta,b}_{-1,{\overline{m}}}(z,{\overline{z}})(z-\beta({\overline{z}}))^{-1}({\overline{z}}-{\overline{b}})^{\overline{m}}$
with linear $\beta\notin{\mathbb{F}}$ (again
$(z-\beta({\overline{z}}))^{-1}\notin\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})$).
We use Lemma 50 to prove that
$g^{\beta,b}_{-1,{\overline{m}}}(z,{\overline{z}})(z-\beta({\overline{z}}))^{-1}({\overline{z}}-{\overline{b}})^{\overline{m}}$
has a single-valued log-Laurent expansion at $\infty$.

By compactness of ${\mathbb{C}}\cup\\{\infty\\}$, every term
$g^{\beta,b}_{-1,{\overline{m}}}(z,{\overline{z}})(z-\beta({\overline{z}}))^{-1}({\overline{z}}-{\overline{b}})^{\overline{m}}$
has a finite number of singular points and hence is in
$\mathcal{S}\mathcal{V}_{\mathbb{C}}$. ∎

######  Example 59.

Proposition 58 does not generalize to all functions
$f\in\mathcal{G}_{\Sigma}$. Consider
$f(z)=\log(z{\overline{z}})/(z-1/{\overline{z}})$ (see Example 48). We have
$f\in\partial_{z}\mathcal{G}_{\\{0,{\overline{z}}^{-1}\\}}^{\mathbb{C}}$ and

$\partial_{z}f(z)=\frac{{\overline{z}}}{z-1/{\overline{z}}}-\frac{{\overline{z}}}{z}-\frac{\log(z{\overline{z}})}{(z-1/{\overline{z}})^{2}}\in\mathcal{G}_{\\{0,{\overline{z}}^{-1}\\}}.$

Expansion at $z=0$ shows that the terms ${\overline{z}}/(z-1/{\overline{z}})$
and $-\log(z{\overline{z}})/(z-1/{\overline{z}})^{2}$ are not in
$\mathcal{S}\mathcal{V}_{\mathbb{C}}$.

######  Corollary 60.

Let $\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$
be finite. Every individual term in the
$\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}$ span of
$\partial_{z}\mathcal{G}_{\Sigma}^{\mathbb{C}}$ (see (68)) is single-valued
(i.e. in $\partial_{z}\mathcal{G}_{\Sigma}^{\mathbb{C}}$).

###### Proof.

Extend $\\{(z-\beta({\overline{z}}))^{-1}\\}$ to a ${\mathbb{C}}$-basis of
$\langle(z-\beta({\overline{z}}))^{-1},\beta({\overline{z}})\in\Sigma({\overline{z}})\rangle_{\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})}$
and use Proposition 58. ∎

###  6.2. The commutative hexagon

######  Lemma 61.

The kernels of $\partial_{z}$ and $\partial_{\overline{z}}$ in
$\mathcal{G}_{\mathbb{F}}$ are $\mathcal{O}_{\mathbb{F}}({\overline{z}})$ and
$\mathcal{O}_{\mathbb{F}}(z)$, respectively. In particular, the kernels of
$\partial_{z}$ and $\partial_{\overline{z}}$ in
$\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ are ${\mathbb{C}}$.

###### Proof.

Let $f\in\mathcal{G}_{\mathbb{F}}$ with $\partial_{z}f=0$. Then, by Theorem 24
(2), we have $f\in\mathcal{H}\mathcal{L}_{\mathbb{F}}({\overline{z}})$.
Because $f\in\mathcal{S}\mathcal{V}_{\mathbb{C}}$ we get from Proposition 8
that $f\in\mathcal{O}_{\mathbb{F}}({\overline{z}})$. For
$f\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ we get
$f\in\mathcal{O}_{\mathbb{F}}({\overline{z}})\cap\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}={\mathbb{C}}$.
The result for the kernel of $\partial_{\overline{z}}$ follows by complex
conjugation. ∎

######  Definition 62.

Let $f\in\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$. Then

$\int_{\mathrm{sv}}f(z)\,\mathrm{d}z=F(z)\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$

if $\partial_{z}F=f$ and $F(0)=0$ (with regularization, see Section 3.2). For
$g\in\partial_{\overline{z}}\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$
we say that a function
$G\in\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ is a
single-valued primitive of $g$, i.e. $G=\int_{\mathrm{sv}}g\,\mathrm{d}z$, if
$\partial_{z}G=g$. We define the complex conjugate
$\int_{\mathrm{sv}}\mathrm{d}{\overline{z}}$ such that complex conjugation
commutes with integration.

Note that by Lemma 61 $\int_{\mathrm{sv}}\mathrm{d}z$ is unique on
$\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ whereas it is multivalued
on $\partial_{\overline{z}}\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$.
Single-valued primitives on
$\partial_{\overline{z}}\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$
differ by anti-analytic functions in
$\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ which, by
Proposition 8, are in $\mathcal{O}_{\mathbb{F}}({\overline{z}})$.

We need the commutativity of the hexagon in Figure 1 to prove the existence of
$\int_{\mathrm{sv}}\mathrm{d}z$ (which will be extended to
$\mathcal{G}_{\mathbb{F}}$ in Definition 72).

######  Theorem 63.

The hexagon in Figure 1 commutes. In particular, for any
$f\in\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ there exists a unique
single-valued primitive
$\int_{\mathrm{sv}}f\mathrm{d}z\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$.

###### Proof.

We first show that the hexagon is well-defined. Let
$f\in\partial_{\overline{z}}\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$.
The primitive $\int_{\mathrm{sv}}f\mathrm{d}z$ is defined up to an anti-
analytic rational function
$g\in\mathcal{O}_{\mathbb{F}}({\overline{z}})\cap\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$.
We get
$g({\overline{z}})=\sum_{a\in{\mathbb{F}}}c_{a}/({\overline{z}}-{\overline{a}})$
with $c_{a}\in{\mathbb{C}}$. This implies $\pi_{\partial_{\overline{z}}}g=0$
which ensures that the hexagon is well-defined.

Now, consider $f\in\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$. We use
induction over the weight $n$ of $f$ in the set $\\{-\infty,0,1,2,\ldots\\}$.