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\def\endproof{$\square$\enddemo} \def\a{\Cal A_} \def\c{\Cal C_}  \def\g{\Cal G_} \def\r{\Cal R_} \def\f{\Cal F_} \def\sy{\text{Sym}^} \def\w{\wedge} 
\def\six{\mathop\sim\limits_6}\def\sgn{\text{sgn}} \def\te#1{P_4\bigl(#1\bigr)}

\topmatter \title Appendix: special values and functional equations of polylogarithms\endtitle
\author Don Zagier\endauthor\endtopmatter    % \def\Bbb{\bold} 

0. Introduction\hfill1\break\indent
1. The basic algebraic relation and the definition of $\a m(F)$\hfill2\break\indent
2. Examples of dilogarithm relations\hfill6\break\indent
3. Examples for higher order polylogarithms\hfill8\break\indent
4. Examples: ladders\hfill10\break\indent
5. Existence of relations among polylogarithm values of arbitrarily high order\hfill12\break\indent
6. A conjecture on linear independence\hfill13\break\indent
7. Functional equations\hfill14\break\indent
References\hfill20\break


\subheading{0. Introduction} In this appendix we want to give a unified explanation of the many linear relations which exist among values of polylogarithms at
algebraic arguments, as well as a framework for understanding the functional equations of these functions. We will content ourselves with describing the
(partially conjectural, partly proved) results, without explaining the reasons which motivate the form the answer takes. This motivation, as well as many more
details of the theoretical setup, can be found in [Z], of which the present appendix (except for \S7, which contains some
new results) is essentially just a summary.

The classical theorem of number theory in which the ``monologarithm" function $\l1(x)=-\log(1-x)$ appears is Dirichlet's theorem, which in a weakened form says:
Let $\a1(F)$ denote the group of units of an algebraic number field $F$ of degree $n=r_1+2r_2$ in the usual notation, $D_F$ the discriminant of $F$. Then 
\item{ }(i) The map $\a1(F)\to\R^n$ which sends $\varepsilon$ to the $n$-tuple of values of $\log|\varepsilon^\sigma|$, $\sigma$ ranging over
the embeddings of $F$ into $\C$, maps $\a1(F)$ to a lattice in an $(r_1+r_2-1)$-dimensional subspace  of $\R^n$; and
\item{}(ii) The covolume of (= volume of a fundamental parallelogram for)
this lattice is a rational multiple of $|D_F|^{1/2}\pi^{-r_2}$ times the residue at $s=1$ of $\zeta_F(s)$, the Dedekind zeta-function of $F$.

\noindent Our goal is to define similarly for each $m>1$ groups $\a m(F)$ such
that (i) the $m$th polylogarithm function defines a map from $\a m(F)$ to a lattice in $\R^n$, and (ii) the covolume of this lattice is up to a simple
factor equal to $\zeta_F(m)$. Assertion (i) implies---since the lattice is finite-dimensional---a 
large number of relations among the polylogarithms with values in $F$, including all known, and conjecturally all, such relations, while (ii)
implies that $\zeta_F(m)$ for any number field $F$ and integer $m>1$ can be expressed in finite terms in terms of values of polylogarithms with values in
$F$. (It was in fact this latter statement, not the existence of relations among polylogarithm values, which was the original motivation for
the conjectures in [Z].) 

The definition of $\a m(F)$ and more details of the above picture are given
in \S1. Several examples in the case of the dilogarithm are discussed in \S2, and for higher polylogarithms in \S3. The reader may want
to look at these sections before reading \S1, which is more theoretical. The relation between the concepts explained here and Lewin's ladders is
discussed in \S4. The following section proves in the case $F=\Q$ that there are enough elements of $\a m(F)$ to produce non-trivial relations, and
in particular that there are non-trivial relations among values of $\l m$ with rational arguments for arbitrary large values of $m$.
In \S6 we state the conjecture that all relations among values of polylogarithms at
algebraic arguments come from specializing functional equations (i.e., are ``analytically derivable" in Lewin's terminology) and hence---since the
processes of verifying or specializing any functional equation are mechanical---in some sense trivial; of course, this is a very non-trivial
use of the work ``trivial," since {\it finding} the functional equations needed is not at all an easy matter. Finally, in \S7 we 
discuss how the same setup as is needed to understand relations among special values of polylogarithms also lets
one search efficiently for functional equations, and report on some new functional equations found by H\. Gangl, including his
spectacular discovery of a functional equation for the hexalogarithm, the first progress beyond $m=5$ in 150 years. 

\subheading{1. The basic algebraic relation and the definition of $\a m(F)$} We are interested in looking for linear dependences of polylogarithm values
at algebraic arguments. It turns out that the only combinations which work are those whose arguments satisfy two conditions. The first condition,  
which is the basic one, is purely algebraic. We will call collections of numbers satisfying this condition {\it good}, or more precisely (since the
condition depends on $m$ and becomes more restrictive as $m$ grows) {\it  good at level $m$}. For the problem of finding functional equations,
discussed in \S7, only the algebraic condition is needed, so the whole problem is to find good combinations of functions of one or several variables
at various levels $m$. The second condition is inductive. At level 2, any ``good" combination of arguments is mapped by the dilogarithm into a certain
lattice in $\R^n$, as sketched in the introductory paragraphs and explained in more detail below, so the group $\a2(F)$ consists of all good combinations and we
automatically get non-trivial relations among dilogarithm values whenever the rank of this group is bigger than the dimension of the lattice to which it maps.
(Actually, its rank is always infinite, as we will see in \S5, so it is merely a question of computation to get as many relations as desired.)
In higher levels $m$, any good combination of arguments has associated to it other combinations which are good at some smaller level $m'$, $2\le m'<m$.
Starting at the bottom ($m'=2$), we require that the image of each of these lower-level combinations under the $m'$-th polylogarithm map is 0 in the
relevant lattice. The group $\a m(F)$, defined as the set of all combinations of elements of $F$ which satisfy this property, then is mapped by the $m$th
polylogarithm map into a certain lattice of finite rank, so we automatically get relations among polylogarithm values if the rank of $\a m(F)$
(which, again, turns out actually to be infinite) is larger than that of the lattice.

We now give the precise definitions. Let $F$ be any field. By a ``combination of elements of $F$" we will mean a formal linear combination $\xi=\sum\limits_in_i\,[x_i]$
where $i$ runs over a finite index set, the $x_i$ are elements of $F^\times$, and the coefficients $n_i$ are integers. (Actually, all our considerations are up to
torsion only, so we could allow $n_i\in\Q$.) For convenience we also allow $[x_i]$ to be 0 or $\infty$ but set $[0]=[\infty]=0$. The set of all combinations $\xi$
forms a group $\f F$, the free abelian group on $F^\times$. An element $\xi\in\f F$ is {\it good at level 2} if it satisfies the relation
$$\sum_i n_i\,[x_i]\w\,[1-x_i]=0\qquad\text{in }\;\Lambda^2(F^\times)\otimes_{\Z}\Q\,,\tag{$*$}$$
i.e. if the sum $\sum n_i[x_i]\w[1-x_i]$ is a torsion element in the exterior square of $F^\times$. (The element $[x_i]\w[1-x_i]$ is to be interpreted as 0 if $x_i=0$, 1
or $\infty$.) Explicitly, ($*$) means that if we pick a basis $p_1$,\dots,\,$p_s$ of the subgroup of $F^\times$ generated by all $x_i$ and $1-x_i$ (modulo torsion) and write
$$x_i=\zeta_i\,\prod_{j=1}^sp_j^{a_{ij}},\qquad1-x_i=\zeta_i'\,\prod_{j=1}^sp_j^{a'_{ij}}$$
where $\zeta_i$, $\zeta'_i\in F^\times$ are roots of unity and $a_{ij}$, $a'_{ij}$ belong to $\Z$, then 
$$\sum_in_i\,\bigl(a_{ij}\,a'_{ik}-a_{ik}\,a'_{ij}\bigr)=0\qquad(1\le j<k\le s).$$
In particular, if for some finite subset $S=\{p_1,\ldots,p_s\}$ of $F^\times$ the set of $x\in F$ for which both $x$ and $1-x$ belong to 
$$\langle S\rangle=\bigl\{\,\zeta\,p_1^{a_1}\ldots p_s^{a_s}\,\mid\,\zeta\in F^\times\text{ a root of unity, }a_1,\ldots,a_s\in\Z\bigr\}$$
has cardinality bigger than $\binom s2$, then we can find good combinations of these elements $x$. A key question, both for finding relations
among special values and for finding functional equations of polylogarithms, will be to find as large sets as possible of (numbers or rational functions) $x\in F$ 
for which all $x$ and $1-x$ belong to a subgroup of $F^\times$ of small rank.

For any vector space, we can identify $\Lambda^2(V)$ with $V\otimes V/\sy2(V)$, where $\sy2(V)$ is the subspace of $V\otimes V$ spanned by elements $x\otimes y+y\otimes x$
(or by elements $x\otimes x$). Thus ($*$) can be interpreted as saying that $\sum n_i[x_i]\otimes[1-x_i]$ belongs to $\sy2(F^\times)$ up to torsion. The generalization
to higher $m$ is to say that a combination $\xi\in\f F$ is {\it good at level $m$} if it satisfies the relation
$$\sum_i n_i\,\underbrace{[x_i]\otimes\ldots\otimes[x_i]}_{m-1}\otimes[1-x_i]\;\in\;\sy m(F^\times)\otimes_{\Z}\Q\,,\tag{$*_m$}$$
where $\sy m(A)$ for any abelian group $A$ is the $\frak S_m$-invariant subspace of the $m$th tensor power $A^{\otimes m}$.
We can write the $(m-1)$-fold tensor product $[x]\otimes\ldots\otimes[x]$ as $[x]^{\otimes(m-1)}$ or more simply, since it automatically belongs to the subspace
$\sy{m-1}(F^\times)$ of $\bigotimes^{m-1}(F^\times)$, as $[x]^{m-1}$. Thus the set $\g m(F)$ of good combinations at level $m$ is the kernel of the linear map
$$\beta_m:\,\f F\;\to\;\bigl((\sy{m-1}(F^\times)\otimes F^\times)\big/\sy m(F^\times)\bigr)\otimes_{\Z}\Q$$ defined on generators by
$$\beta_m\bigl([x]\bigr)\;=\,\cases[x]^{m-1}\otimes[1-x]\;\pmod{\sy m(F^\times)}&\quad\text{if $x\ne0,\,1,\,\infty$}\,,\\\phantom{[x]^{m-1}}0&\quad\text{if
$x=0,\,1,\,\infty$}\,.\endcases$$ We can also write $\g m(F)$ in a way more analogous to the definition for $m=2$ as the kernel of the map
$[x]\mapsto[x]^{m-2}\otimes\bigl([x]\w[1-x]\bigr)\in \sy{m-1}(F^\times)\otimes\Lambda^2(F^\times)$, since for any vector space $V$ there is a natural injection
$\sy{m-1}(V)\otimes V/\sy m(V)\to\sy{m-2}(V)\otimes\Lambda^2(V)$.

The first relation between $\g m(F)$ and the polylogarithm function is that functional equations of $\l m$ modulo lower order polylogarithms are given by combinations
$\xi=\sum n_i[x_i]$ of rational functions $x_i=x_i(t)$ satisfying $(*_m)$. Since $\l m$ cannot be extended to a continuous one-valued function on the whole complex plane,
we will work instead with the modified function $$P_m(x)=\Re_m\biggl(\sum_{j=0}^{m-1}\frac{2^j\,B_j}{j!}\,\bigl(\log|x|\bigr)^j\,\l{m-j}(x)\biggr)\,;$$
here $\Re_m$ denotes $\Im$ or $\Re$ according as $m$ is even or odd and $B_j$ the $j$th Bernoulli number ($B_0=1$, $B_1=-\frac12$, $B_2=\frac16$, $B_3=0$, \dots).
This function is continuous and real-valued on $\P=\C\cup\{\infty\}$ and is real-analytic except at $0$, 1 and $\infty$. (The function $P_2(x)$ is just
the Bloch-Wigner function $D(x)$ defined in 1.5.1 of this book.)  Using $P_m$ instead of $\l m$ also has
the advantage that the lower order terms in all functional equations cancel, so that $\sum_iP_m\bigl(x_i(t)\bigr)=$constant for any good combination 
$\sum n_i[x_i(t)]$. The proof, which is not difficult, is given in \S7 of [Z] and will not be repeated here. As simple examples, we note that
 $$\beta_m\bigl(\frac1x\bigr)=\bigl(-[x]\bigr)^{m-1}\otimes\bigl([-1]+[1-x]-[x]\bigr)=(-1)^{m-1}\beta_m(x)$$ for any $x\in F$,
since $[-1]$ is a torsion element of $F^\times$ and $[x]^{m-1}\otimes[x]$ belongs to $\sy m(F^\times)$, and similarly
 $$\beta_m(x^2)=\bigl(2\,[x]\bigr)^{m-1}\otimes\bigl([1-x]+[1+x]\bigr)=2^{m-1}\,\bigl(\beta_m(x)+\beta_m(-x)\bigr)$$ 
or more generally $\beta_m(x^N)=\sum_{\zeta^N=1}\beta_m(\zeta x)$ for any $N\ge1$, corresponding to the inversion and distribution relations
$$P_m(1/x)=(-1)^{m-1}P_m(x),\qquad P_m(x^N)=N^{m-1}\sum_{\zeta^N=1}P_m(\zeta x)$$
of the polylogarithm. Many other examples will be discussed in \S7.

We now return to number fields and the definition of $\a m(F)$. If $\phi$ is any homomorphism from $F^\times$ to $\Z$ (e.g\. $\phi=v_p$ for $F=\Q$, where $p$ is
a prime number and $v_p(x)$ denotes the power of $p$ contained in a rational number $x$), then for $m>2$ the map $\iota_\phi:\f F\to\f F$ sending $[x]$ to
$\phi(x)\,[x]$ sends $\g m(F)$ to $\g{m-1}(F)$ (because $\beta_{m-1}\circ\iota_\phi=(\phi\otimes\text{id}^{\otimes(m-1)})\circ\beta_m$  under the natural
identification of $\Z\otimes(F^\times)^{\otimes(m-1)}$ with $(F^\times)^{\otimes(m-1)}$). If we have $r$ homomorphisms $\phi_i$ ($1\le i\le r$), then for $m\ge r+2$ the
composite map $\iota_{\phi_1}\circ\ldots\circ\iota_{\phi_r}:\g m(F)\to\g{m-r}(F)$ is independent of the order of the $\phi_i$ and will be denoted simply
$\iota_{\phi_1}\ldots\iota_{\phi_r}$. The elements $\iota_{\phi_1}\ldots\iota_{\phi_r}(\xi)\in\g{m-r}(F)$ ($1\le r\le m-2$) for an element $\xi$ of $\g m(F)$ will be
called the good combinations {\it associated to} $\xi$. Up to linear combinations there are only finitely many of them, since only the restrictions of the $\phi_i$
to the subgroup of $F^\times$ generated by the elements occurring in $\xi$ are important and we can let $\phi_i$ run over a basis of the group of homomorphisms from
this group to $\Z$.

The definition of $\a m(F)$ is now as follows. We start with $\a2(F)=\g2(F)$. For $\xi=\sum n_i[x_i]\in\a2(F)$ and each embedding $\sigma:F\into\C$, we consider
$P_2(\xi^\sigma)$, where $\xi^\sigma$ denotes $\sum n_i[x_i^\sigma]$ and $P_2(\xi^\sigma)$ denotes $\sum n_i P_2(x_i^\sigma)$ (we shall use this abbreviated notation
for the value of functions on combinations of arguments from now on). There are only $r_2$ essentially distinct such values, since the relation $P_2(\overline x)=-P_2(x)$
for $x\in\C$ implies that $P_2(\xi)$ vanishes for the real embeddings of $F$ and is the same up to sign for each conjugate pair of complex embeddings. Thus, picking
one of each conjugate pair of non-real embeddings, we get a map $\vec P_2$ from $\a2(F)$ to $\R^{r_2}$ by sending $\xi$ to $\{P_2(\xi^\sigma)\}_\sigma$.
The image $\vec P_2(\a2(F))$ is contained in a {\it lattice} (= discrete subgroup of maximal rank) $\r2=\r2(F)\subset\R^{r_2}$.  
Therefore, if $\a2(F)$ has rank $>r_2$ (and we will see later that its rank is actually infinite), then the kernel $\c2(F)$ of $\vec P_2$ contains
non-zero elements and we get non-trivial relations among dilogarithms of complex algebraic arguments. Furthermore, if $\xi\in\c2(F)$ and $\sigma:F\into\R$ is
a real embedding of $F$, then $L_2(\xi^\sigma)$, where $L_2(x)=\l2(x)+\frac12\log|x|\log|1-x|$ ($x\in\R$) is the Rogers dilogarithm function (cf\. 1.2.3), 
is a rational multiple of $\pi^2$, so we also get relations among dilogarithms of real algebraic arguments.
For $m=3$ we define $\a3(F)$ as the subset of $\g3(F)$ consisting of all good combinations $\xi$ for which $\iota_\phi(\xi)\in\a2(F)$ belongs to $\c2(F)$ for
every homomorphism $\phi:F^\times\to\Z$. This condition can be checked by numerical computation  since the element $P_2(\iota_\phi(\xi))$ belongs to the discrete group $\r2$ 
and hence is recognizably zero or non-zero. For $\xi\in\a3(F)$ we consider the collection of all $P_3(\xi^\sigma)$ where (since $P_3(x)$ is invariant rather than
anti-invariant under complex conjugation of $x$) $\sigma$ now ranges over the real and half the complex embeddings of $F$. This defines a map $\vec P_3:\a3(F)\to\R^{r_1+r_2}$ 
whose image is again contained in a lattice $\r3\subset\R^{r_1+r_2}$, so again we get a non-trivial kernel $\c3(F)$ and non-trivial 
relations among polylogarithm values. Similarly, for $m>2$  we consider the map $\vec P_m:\f F\to\R^{n_\mp}$, where
$\pm1=(-1)^m$, $n_+$ and $n_-$ denote $r_1+r_2$ and $r_2$ respectively, and $\vec P_m(\xi)$ is the collection of $P_m(\xi^\sigma)$ with $\sigma$ ranging over half the complex
embeddings ($m$ even) or all the real embeddings and half the complex embeddings ($m$ odd). The subgroup $\a m(F)\subseteq\g m(F)\subset\f F$ will be defined inductively
in such a way that its image under $\vec P_m$ is contained in a certain lattice $\r m=\r m(F)\subset\R^{n_\mp}$, in which case the kernel
$$\c m(F)=\bigl\{\xi\in\a m(F)\;\mid\;P_m(\xi^\sigma)=0\quad\text{for all $\sigma:F\into\C$}\,\bigr\}$$ is a subgroup of corank at most $n_\mp$. Assuming that $\a{m'}(F)$ and $\c{m'}(F)$
have been defined for $m'<m$, we define $\a m(F)$ by 
$$\a m(F)=\bigl\{\xi\in\g m(F)\;\mid\;\iota_\phi(\xi)\in\c{m-1}(F)\quad\text{for all homomorphisms $\phi:F^\times\to\Z$}\,\bigr\}\,.$$
Explicitly, this means that starting with $\xi\in\g m(F)$, we verify $\xi\in\a m(F)$ by checking first that all associated elements 
$\iota_{\phi_1}\ldots\iota_{\phi_{m-2}}(\xi)\in\a2(F)$ map to 0 under $\vec P_2$, then that all associated elements
$\iota_{\phi_1}\ldots\iota_{\phi_{m-3}}(\xi)$ (which are then known to belong to $\a3$) map to 0 under $\vec P_3$, and so on in succession for
all associated elements $\iota_{\phi_1}\ldots\iota_{\phi_r}(\xi)$ for $r=m-2$, $m-3$, \dots, 2, 1.

We do not give any further explanation of these definitions here; the examples in the following sections should make it clear how they work in practice, while the background
and motivating ideas are explained in more detail in [Z]. However, we should say a few words about the relationship to $K$-theory and about what has been proved so far and
by whom. The former can be understood without needing to know the definition of algebraic $K$-groups $K_n(F)$. One need only know that there are such groups and that
by a result of Borel [B], the even-index groups are finite while the odd-index groups $K_{2m-1}(F)$ ($m>1$) are mapped isomorphically (up to torsion) onto a lattice
in $\R^{n_\mp}$ by the so-called ``regulator mapping." The lattice $\r m(F)$ we have been speaking about is the image of the regulator mapping (or more precisely, any
lattice commensurable with this image, since everything we are saying is known only up to groups of finite order). Borel also showed that the covolume of $\r m\subset\R^{n_\mp}$
is a rational multiple of $|D_F|^{1/2}\,\zeta_F(m)/\pi^{mn_\pm}$. Hence if the map $\vec P_m:\a m(F)\to\r m$ is surjective, then $\zeta_F(m)$ can be expressed in
terms of polylogarithms of order $m$ with arguments in $F$.

The relationship between the dilogarithm and the regulator lattice $\r2$ was found by Bloch (for this and later
developments concerning $K_3$ and dilogarithms, see Suslin's ICM talk [S]). In this case the map $\vec P_2:\a2\to\r2$ is surjective (after tensoring with $\Q$)
and its kernel is the subgroup of $\a2(F)$ spanned by the 5-term functional equation of the dilogarithm. 
The above picture for general $m$ was formulated conjecturally in [Z]. The case $m=3$ was proved completely
by Goncharov [G], who showed that the map $\vec P_3:\a3(F)\to\R^{n_+}$ maps not only into, but onto the regulator lattice $\r3$ (at least after tensoring with $\Q$)
and also gave a complete description of the kernel in terms of a new 22-term functional equation for the trilogarithm.
The fact that $\vec P_m$ maps $\a m(F)$ into the Borel regulator lattice for arbitrary $m$ was proved by Deligne and by Beilinson for cyclotomic
fields and by Beilinson in the general case [BD].  For $m>3$ the surjectivity is not known except in the cyclotomic case, so that the desired corollary that $\zeta_F(m)$ 
can be expressed in terms of  special values of polylogarithms at arguments in $F$ is still a conjecture in general. We also conjecture that $\c m(F)$ can be described
completely in terms of the functional equations of the $m$th polylogarithm, but this also is known only for $m=2$ and 3.

Finally, we should mention that the generalized Rogers function $L_m(x)$ ($x\in\R$) defined in 1.5.1 agrees with $P_m$ for $m=3$
and $x$ real, while for $m>3$ odd and $x$ real it is equal to $\sum_{r=0}^{m-2}(\log|x|)^r\,P_{m-r}(x)/(r+1)!$ (only the terms with $r$ even in this sum contribute,
since $P_{m'}(x)=0$ for $m'$ even and $x$ real). Thus for elements $\xi\in\a m(F)$ ($m$ odd) and real embeddings $\sigma$ of $F$, the elements $P_m(\xi^\sigma)$ and
$L_m(\xi^\sigma)$ agree (because each term $\sum n_i(\log|x_i|)^rP_{m-r}(x_i)$, $r>0$, vanishes by virtue of the inductive definition of $\a m$), and therefore our
relations among values of $P_m(x)$ at real algebraic arguments can be reinterpreted as relations among the same values of $L_m(x)$. For $m$ even, $P_m(x)$ vanishes
identically for $x$ real, but if we take an element $\xi\in\a m(F)$ for which $P_m(\xi^\sigma)=0$ for all complex embeddings $\sigma$, then it is apparently a
consequence of the results in [BD] that each $L_m(\xi^\sigma)$ ($\sigma:F\into\R$) is a rational multiple of $\pi^m$.


\subheading{2. Examples of dilogarithm relations} For $m=2$ there is no distinction between $\g m$ and $\a m$, so we just have
to look for good combinations at level 2, i.e., combinations $\xi=\sum n_i[x_i]$ satisfying ($*$). For such $\xi$, $P_2(\xi)=D(\xi)$
belongs to the $r_2$-dimensional lattice $\r2$. Thus if we find $r_2$ elements $\xi$ with linearly independent images, we obtain
$\zeta_F(2)$ as $\pi^{2(r_1+r_2)}/\sqrt{|D_F|}$ times a rational number times an $r_2\times r_2$ determinant of integral
linear combinations of dilogarithm values, and if we have more than $r_2$ good combinations $\xi$, then they will have linearly
dependent images in the lattice and we obtain linear relations over $\Q$ among the values $D(x_i)$ (resp\. $L_2(x_i^\sigma)$ modulo $\pi^2$ for
the real embeddings $\sigma$ of $F$, where $L_2(x)$ is the Rogers dilogarithm). 

As an example over $\Q$, take elements $x\in\Q$ such that $x$ and $1-x$ contain no primes except 2, 3, 5 and 7. There are exactly
375 such $x$, forming 63 orbits under the group generated by $x\mapsto1/x$, $x\mapsto1-x$. For each one, $\beta_2(x)=[x]\w[1-x]$ is a linear combination
of the six elements $$[2]\w[3],\quad[2]\w[5],\quad[2]\w[7],\quad[3]\w[5],\quad[3]\w[7],\quad[5]\w[7],$$
so we get 57 essentially different linearly independent linear combinations $\xi$ belonging to $\a2(\Q)$, for each of which
$L_2(\xi)$ is a rational multiple of $\pi^2$. For instance, if we pick at random the seven elements
$\dsize\frac13$, $\dsize-\frac16$, $\dsize\frac27$, $\dsize\frac18$, $\dsize\frac19$, $\dsize\frac1{21}$, and $\dsize\frac1{28}$ (any other seven
would do just as well), then since $7>6$ we must find at least one non-trivial element of $\a2(\Q)$. In fact, calculating
$$\beta_2(\frac13)=[\frac13]\w[\frac23]=[2]\w[3],\quad\beta_2(-\frac16)=[-\frac16]\w[\frac76]=-[2]\w[7]-[3]\w[7],\quad\text{etc\.},$$
we find that two linearly independent combinations of the 7 elements in question are annihilated by $\beta_2$, namely
$6\bigl[\frac13\bigr]-\bigl[\frac19\bigr]$ and $3\bigl[-\frac16\bigr]-\bigl[\frac18\bigr]+\bigl[\frac19\bigr]+\bigl[\frac1{28}\bigr]$. The image of each
of these under the Rogers dilogarithm  must be a rational multiple of $\pi^2$, and indeed, 
$$6\,L_2\bigl(\frac13\bigr)\,-\,L_2\bigl(\frac19\bigr)\,=\,\frac{\pi^2}3\,,\qquad
3\,L_2\bigl(-\frac16\bigr)\,-\,L_2\bigl(\frac18\bigr)\,+\,L_2\bigl(\frac19\bigr)\,+\,L_2\bigl(\frac1{28}\bigr)\,=\,-\frac{\pi^2}{12}\,.$$

Taking instead a field with $r_2>0$, we let $F=\Q(\sqrt{-7})$. Then for the two elements $x_1=(1+\sqrt{-7})/2$ and $x_2=(-1+\sqrt{-7})/4$, both
$1-x_1$ and $1-x_2$ belong to the group generated by $-1$, $x_1$, and $x_2$, so both $\beta(x_1)$ and $\beta(x_2)$ are multiples of $[x_1]\w[x_2]$.
The multiples turn out to be 1 and $-2$, respectively, so $2[x_1]+[x_2]$ belongs to $\a2(F)$. Hence $2D(x_1)+D(x_2)$ must be a rational multiple
of $\zeta_F(2)/\pi^2\sqrt7$, and indeed one finds 
$$2D\bigl(\frac{1+\sqrt{-7}}2\bigr)+D\bigl(\frac{-1+\sqrt{-7}}4\bigr)=\frac{21\sqrt7}{4\pi^2}\,\zeta_{\Q(\sqrt{-7})}(2).$$

As a third example, we consider the number $u=\sqrt\rho$, where $\rho=(\sqrt{5}-1)/2$ (cf\. Chapter 5, section 5.2.1).
Here the relation (5.4) implies that $\xi_0=[u^6]-4[u^3]+6[u]$ is a good element of $\f F$, $F=\Q(u)$, since
$$\align\beta_2(\xi_0)&=[u^6]\w[1-u^6]-4[u^3]\w[1-u^3]+6[u]\w[1-u]\\&=6[u]\w[(1-u^6)(1-u^3)^{-2}(1-u)]=12[u]\w[u]=0.\endalign$$
However, $\xi_0$ does not belong to the subgroup $\c2(F)$ of $\a2(F)$, because, denoting by $\sigma$ the embedding of $F$ into $\C$ which sends $u$ to
$u^\sigma=i/u$, we have $D(\xi_0^\sigma)=8.6124152\ldots\ne0$. Therefore we should not expect
$L_2(\xi_0)/\pi^2$, which is the number given by (5.6), to be rational. What we {\it do} expect is that $D(\xi_0^\sigma)$ is a rational multiple of $\pi^{-6}\zeta_F(2)$, 
since for the field $F$ we have $r_1=2$, $r_2=1$, $D_F=400$. To compute numerically, we decompose $\zeta_F(s)$ as $\zeta_{F_1}(s)L_{F_1}(s)$ where $\zeta_{F_1}(s)$ is the Dedekind
zeta function of $F_1=\Q(\sqrt5)$ and 
$$L_{F_1}(s)=\frac1{1+5^{-s}}\prod_{p\equiv11,\,13,\,17,\,19}\frac1{1-p^{-2s}}\prod_{p\equiv3,\,7}\frac1{1+p^{-2s}}\prod_{p\equiv1,\,9}\bigl(\frac1{1+\varepsilon_pp^{-s}}\bigr)^2$$
(here the congruences on primes $p$ are modulo 20 and $\varepsilon_p=\bigl(\dsize\frac \rho p\bigr)=\bigl(\dsize\frac{2i+1}p\bigr)$ for $p\equiv1$ or 9 mod 20, where $\rho$ and $i$
are solutions of $\rho^2+\rho-1\equiv i^2+i\equiv0\pmod p$). We have $\zeta_{F_1}(2)=20\pi^4/75\sqrt5$ and (computing numerically with the Euler product, using all primes up to
2500) $20^{3/2}\pi^{-2}L_{F_1}(2)\approx8.612406$, agreeing with $D(\xi^\sigma_0)$ to the accuracy of the computation. If we want to find elements of $\c2(F)$ and hence
relations among the Rogers dilogarithms of elements of $F$, then we must look at combinations involving other numbers of $F$ than just powers of $u$, i.e., it is not enough to
consider ladders only. For instance, the group of units of $F$ is generated (up to torsion) by $u$ and $v=1-u$, and we can look at elements $x\in F$ for which both $x$ and $1-x$
belong to this group. Up to equivalence by the group generated by $x\mapsto1/x$ and $x\mapsto1-x$, under which $D(x)$ is invariant up to sign, there are 5 such elements,
namely $x=u$, $-u$, $u^2$, $u^2v$ and $v^2/u^3$. For these elements $1-x$ equals $v$, $u^4/v$, $u^4$, $u^7/v$ and $v/u^6$, respectively, so $\beta_2[x]=[x]\w[1-x]$
equals $[u]\w[v]$ times 1, $-1$, 0, $-9$ and 9, respectively. We therefore get four linearly independent elements $[u]+[-u]$, $[u^2]$, $[u^2v]+9[u]$ and
$[u^2v]+[v^2/u^3]$ belonging to $\a2(F)$. The first two of these are proportional by the duplication formula and
uninteresting because they reduce to relations from the smaller field $F_1$. Computing numerically, we find that 
$D\bigl((u^2v)^\sigma\bigr)+9D\bigl(u^\sigma\bigr)$ equals $D(\xi_0^\sigma)$ but that $D\bigl((u^2v)^\sigma\bigr)+D\bigl((v^2/u^3)^\sigma\bigr)$ vanishes. Hence
$[u^2v]+[v^2/u^3]$ and $[u^2v]+9[u]-\xi_0$ belong to $\c2(F)$ and should map under the Rogers dilogarithm to rational multiples of $\pi^2$, and indeed,we have
 $$L_2(u^2v)+L_2(v^2/u^3)=\frac1{20}\,\pi^2\,,\qquad L_2(u^2v)+9L_2(u)-L_2(\xi_0)=\frac{41}{60}\,\pi^2\,.$$ 


\subheading{3. Examples for higher order polylogarithms} Many examples of relations among trilogarithms of algebraic arguments are given
in \S\S3--5 of [Z] as motivation for the form of the conjectures explained in \S1. Here we give instead
two examples for the field $F=\Q$ (also both taken from [Z]). The first illustrates the necessity of the extra condition in the definition of $\a m$ as
opposed to $\g m$, while the second gives an example of a relation for heptalogarithms, beyond the range of known functional equations.

For the first example, we proceed as in the example for $m=2$ in \S3, but using only the primes 2 and 3 (if we used 2, 3, 5 and 7 again,
we would get many more examples). There are 21 numbers $x\in\Q$ for which $x$ and $1-x$ contain only the prime factors 2 and 3, but this
number is cut down to 11 if we do not take both $x$ and $x^{-1}$ (for which the values of all polylogarithms are the same up to sign), and
further cut down to 8 if we eliminate the numbers $-1$, 1/4 and 1/9 for which the polylogarithm reduces to simpler values by virtue of the duplication
equation. These values are 1/2, 1/3, 2/3, $-1/2$, 3/4, $-1/3$, 8/9 and $-1/8$. The image of each of them under $\beta_5$ belongs to a space of
dimension 4 (this is the dimension of $\sy4(V)\otimes V/\sy5(V)$ for $V$ the 2-dimensional subspace of $\Q^\times$ spanned by 2 and 3), so
we must have at least four independent elements $\xi\in\text{Ker}(\beta_5)$, and indeed there are exactly four, namely $[1/2]$, $[-1/3]-2[1/3]$,
$[-1/8]-162[-1/2]$ and $[8/9]-9[3/4]-36[2/3]-18[-1/2]-6[1/3]$. This gives a 4-dimensional subspace of $\g5(\Q)$. However, to get elements
of $\a5(\Q)$, we need three further conditions, namely that the images of $\xi$ in $\a3(\Q)$ under the three maps sending $[x]$ to $v_2(x)^2[x]$,
$v_2(x)v_3(x)[x]$, and $v_3(x)^2[x]$ all map to zero under the trilogarithm map $\vec P_3:\a3(\Q)\to\Q\zeta(3)$. This cuts down the dimension from four
to one, the unique surviving relation being $\xi=[-1/8]-126[1/2]-162[-1/2]$. This element should therefore map to a rational multiple of $\zeta(5)$
under $P_5$ or $L_5$, and indeed we find that $P_5(\xi)=L_5(\xi)=\frac{403}{16}\,\zeta(5)\,$.

To get an example for the heptalogarithm takes more work.  If we consider the set of all $x$ such that both $x$ and $1-x$ contain only the first $s$ primes for 
some $s$ (as we did for $m=2$ with $s=4$ and for $m=5$ with $s=2$), then the number of conditions we have to satisfy is so large that the first value of
$s$ for which there are enough $x$ to give a non-trivial element of $\a7(\Q)$ is 8 (i.e., use all primes less than 20), for which there are 10946 elements $x$
(up to inversion) and ``only" 10662 conditions to be satisfied (cf\. [Z], \S10A). This system of equations is far too large to solve numerically. Instead we
consider $x\in\Q$ for which $x\in\langle\{2,\,3\}\rangle$ and $1-x\in\langle\{2,\,3,\,5,\,7\}\rangle$. There are 29 such $x$ (taking only one of each pair
$x$, $1/x$ and omitting squares as before). On the other hand, there are 28 conditions to be fulfilled: first 20 to get good combinations at level 7 (if $V$ and
$W$ are the 2- and 4-dimensional subspaces of $\Q^\times\otimes_{\Z}\Q$ generated by $\{2,\,3\}$ and $\{2,\,3,\,5,\,7\}$, respectively, the number of
conditions is $\dim(\sy6(V)\otimes W/\sy7(V))=\dsize\binom76\cdot 4-\binom87$) and then a further 8 to ensure that each of the combinations $\iota_{v_2}^j\iota_{v_3}^{r-j}(\xi)$
($r=4$ or 2, $0\le j\le r$) maps to 0 in the one-dimensional lattice $\r{7-r}$. Since 29 is bigger than 28, we must find a solution of this system of equations, i.e., a
combination $\xi\in\a7(\Q)$. This solution turns out to be unique up to multiplication by a constant and is given by $\xi=\sum n_i[x_i]\in\a7(\Q)$, where the $n_i$ 
(normalized for convenience to be in $\frac16\Z$ rather than $\Z$) and $x_i$ are given by the following table:
$$ \vbox{\tabskip=0pt \offinterlineskip
\halign{ \tabskip=1em plus 2em\hfill # &#\hfill &#\phantom{x}&
 \hfill #&#\hfill &#\phantom{x} &\hfill   #&#\hfill\tabskip=0pt\cr
\noalign{\vskip5pt}$n_i$\phantom{xx}& \phantom{xx}$x_i$ && $n_i$\phantom{xx}& \phantom{xx}$x_i$&&$n_i$\phantom{xx}&\phantom{xx}$x_i$\cr
\multispan2\hrulefill&&\multispan2\hrulefill &&\multispan2\hrulefill \cr \noalign{\vskip4pt}
$-25111753072$&$\phantom-$1/3&&$-284585110$&$\phantom-$1/8&&1765911&$\phantom-$3/128\cr
$-27461584367$&$-1/3$&&470985412/3&$-1/8$&&478706760&$\phantom-$2/9\cr
$-171330250$&$-1/9$&&38987641&$-1/24$&&$-66158750$&$\phantom-$8/9\cr
57577037&$-1/27$&&17015061/2&$-1/48$&&15912813&$-9/16$\cr
$-151540388696$&$\phantom-$1/2&&$-11528187258$&$\phantom-$2/3&&23786119&$\phantom-$2/27\cr
$-136446322032$&$-1/2$&&$-6563312469$&$-2/3$&&$-2879429$&$-8/27$\cr
$-2209899405$&$\phantom-$1/6&&2802854628&$\phantom-$3/4&&2585366&$\phantom-$27/32\cr
$-2199243270$&$-1/6$&&$-751304106$&$-3/4$&&111363&$\phantom-$32/81\cr
43524&$-1/4374$&&$-785318380$&$\phantom-$3/8&&$-2372265$&$-2/243$\cr
$-7089743800$&$-1/4$&&11883921&$-3/32$&&  &  \cr }}$$
The theory now predicts that $\sum n_iP_7(x_i)$ should be a rational multiple of $\zeta(7)$, and indeed, computing numerically we find that it equals
$-\dsize\frac{1020149599795}{96}\,\zeta(7)\,$ to high precision.


\subheading{4. Examples: ladders} We now come to the subject of Leonard Lewin's ladders (note once again the fascination with the letter 
``L" which marks this field; cf\. [L], p\. 191), the source of most of the examples in the book, and show how they fit into the theory sketched so far.
Briefly, ladders are the special case of combinations $\xi=\sum n_i[x_i]$ in which all of the $x_i$ are  powers of a single number $\alpha$. The advantage of
making this restriction is that the conditions needed to make $\xi$ belong to $\a m$ become much more transparent and easier to check (and to fulfill) than
in the general case. The disadvantage, of course, is that it is very difficult and requires great ingenuity to produce examples, whereas providing examples
of general elements in $\g m$ or $\a m$ is something which can be done in a mechanical manner. In particular, while we can show (\S5) that there exist polylogarithmic
relations for arbitrarily high orders $m$, it is not at all clear---indeed, rather unlikely---that there are any valid ladders at all with $m$ larger than, say, 20.

Suppose, then, that all of the $x_i$ are powers of a single number $\alpha$. Since $[x]$ and $[1/x]$ are essentially equivalent for all polylogarithmic
purposes, we can restrict to positive powers. Then the combinations $\xi$ we are looking for can be written as 
$$\xi=\sum_{j=1}^\infty n_j\,[\alpha^j]\qquad(n_j\in\Z,\quad n_j=0\text{ for all but finitely many $j$})\,.\tag 1$$
(From now on we will abbreviate the condition in brackets as $\,\{n_j\}\in\mathop\oplus\limits_{j\ge1}\Z$.) 
If we compute the image of this under the map $\beta_m$, we find (since $[\alpha^j]=j[\alpha]$ in $F^\times$)
$$\beta_m(\xi)=\sum_{j\ge1}j^{m-1}\,n_j\,\bigl[\alpha]^{m-1}\otimes[1-\alpha^j\bigr]
=\bigl[\alpha\bigr]^{m-1}\otimes\bigl[\prod_{j\ge1}(1-\alpha^j)^{j^{m-1}n_j}\bigr].$$
This is 0 modulo $\sy m(\langle\alpha\rangle)$ if and only if $\prod_j(1-\alpha^j)^{j^{m-1}n_j}$ is (a root of unity times) a power of $\alpha$. In other words,
turning things around, given any {\it cyclotomic relation}
$$\prod_{j=1}^\infty\bigl(1-\alpha^j\bigr)^{b_j}\;=\;(\text{root of 1})\,\times\,\alpha^N\qquad(\{b_j\}\in\mathop\oplus\limits_{j\ge1}\Z,\;N\in\Z)\tag 2$$
we get good combinations $\sum\limits_{j=1}^\infty j^{-m+1}\,b_j\,[\alpha^j]\,$ for {\it all levels} $m$. (Of course, if we want integral combinations, we must
multiply these by $J^{m-1}$, where $J$ is a common multiple of the $j$ with $n_j\ne0$.) This is the beauty of ladders: they give an immediate construction of
elements of $\g m(F)$ for all $m$ simultaneously. However, to get relations among polylogarithm values, we need combinations in $\a m$, not $\g m$. Here, too,
the special property that $V=\langle\{x_i\}\rangle$ is 1-dimensional simplifies life considerably, since it means that there is up to scalar factors only
one homomorphism $\phi$ from $V$ to $\Q$ (namely, the one sending $\alpha^j$ to $j$) and hence only one way to associate to a good combination
$\xi=\sum n_j[\alpha^j]$ of order $m$ good combinations of smaller order $m'=m-r$, namely $\xi'=\iota_\phi ^r(\xi)=\sum_j j^rn_j[\alpha^j]$. Thus we get
the following inductive picture.  Let $\{b_j^{(1)}\}$, \dots, $\{b_j^{(d)}\}$ be a multiplicatively independent set of cyclotomic relations (2) for the
same number $\alpha$. Then for each $m$ we get $d$ linearly independent elements $\xi_m^{(\nu)}=\sum j^{-m}\,b_j^{(\nu)}\,[\alpha^j]\,$ of $\g m(F)$, where
$F=\Q(\alpha)$. For $m=2$ each of these elements belongs to $\a 2$ and maps to the $r_2$-dimensional lattice $\r2$, so we get at least $d-r_2$ dilogarithm
relations. Changing our basis for the set of cyclotomic relations we can assume that these are $\xi_2^{(1)}$, \dots, $\xi_2^{(d-r_2)}$, i.e., for each
$\nu\le d-r_2$ we have $\sum j^{-2}b_j^{(\nu)}D\bigl((\alpha^\sigma)^j\bigr)=0$ if $\alpha^\sigma$ is complex, 
$\sum j^{-2}b_j^{(\nu)}L_2\bigl((\alpha^\sigma)^j\bigr)\in\Q\,\pi^2$ if $\alpha^\sigma$ is real, and $\xi_3^{(\nu)}=\sum j^{-3}b_j^{(\nu)}[\alpha^j]\in\a3(F)$.
Now the elements $\xi_3^{(1)}$, \dots, $\xi_3^{(d-r_2)}$ map under $D_3$ to the $(r_1+r_2)$-dimensional lattice $\r3$, so we get at least $d-r_2-(r_1+r_2)=d-n$ ($n=[F:\Q]$)
linearly independent relations among the trilogarithms of the $\alpha^j$ and the same number of elements in the next higher group $\a4(F)$. Continuing in this\
way, we find that the dimension goes down alternately by $r_2$ and $r_1+r_2$, so by $n$ every two steps. Hence after approximately $\dsize\frac{2d}n$
steps the process terminates, unless we are very lucky (i.e., there happen to be more linear relations than is forced by the number of equations to be satisfied, a rather
rare occurrence) or $d$ is infinite. In fact the second alternative cannot happen, since one can show that the number 
$$d(\alpha)\mathop=_{\text{DEF}}\;\text{rk}_{\Z}\bigl\{\{b_j\}\in\mathop\oplus\limits_{j\ge1}\Z\;\mid\;\prod_{j=1}^\infty(1-\alpha^j)^{b_j}\in\langle\alpha\rangle\bigr\}$$
is finite for any algebraic number $\alpha$ which is not a root of unity (of course $d(\alpha)=0$ if $\alpha$ is transcendental).

We now illustrate all of this with two examples. We first take $\alpha$ to be the number $\omega$ treated in [AL] and in Chapter 4 of the present book, i.e.,the
root of $\omega^3=\omega+1$. Here $d(\alpha)$ is (at least) 12, the corresponding {\it special} exponents ($=\,$exponents $j$ for which a positive power of
$1-\alpha^j$ belongs to the subgroup  generated by $\alpha$ and $1-\alpha^i$ with $i<j$) being 1, 2, 3, 5, 8, 12, 14, 18, 20, 28, 30 and 42. Since $r_1=r_2=1$,
 we can go up to $m=8$, the number of linear relations obtained for the $m$th order polylogarithms being 11, 9, 8, 6, 5, 3, 2  for $m=2$, 3, \dots, 8, respectively.
At the 9th level there are no more relations. However, if
 we include 1 as well as the positive powers of $\omega$ then there is one more relation in odd levels 
and in particular we get an expression of $\zeta(9)$ as a rational linear combination of $D_9(\omega^n)$ with $n>0$. More details of this example can be
found in Chapter 4 and in \S9C of [Z].

The most spectacular example is the Salem number treated in Chapter 16 (section 16.3), namely the solution of 
$\alpha^{10}+\alpha^9-\alpha^7-\alpha^6-\alpha^5-\alpha^4-\alpha^3+\alpha+1=0$.  
Here $d(\alpha)$ is (at least, and probably exactly) 71, the special exponents ranging up to $j=360$.  Here $r_2=4$ and $r_1+r_2=6$, so the number
of conditions to be satisfied at the even and odd steps is a priori 4 and 6, respectively, but because the two real conjugates of $\alpha$ are inverses of
one another and $D_m(x^{-1})=D_m(x)$ for $m$ odd, there are in fact only 5 independent conditions at odd levels. We therefore get successively 
(at least) 71, 67, 62, 58, \dots elements of $\r2$, $\r3$, $\r4$, $\r5$, \dots. Thus the ladder reaches up to $m=16$. 

By the way, to find the special exponents, one proceeds as
follows. First calculate the norm of $\alpha^j-1$ by multiplying the conjugates numerically for $j$ up to, say, 1000.  The norm is 1 for 
the 22 values $j=1$, 2, 3, 5, 6, 7, 9, 10, 11, 13, 17, 18, 21, 23, 27, 29, 34, 37, 47, 63, 65 and 74. 
Since the unit rank is $r_1+r_2-1=5$, we lose 4 relations, because $\alpha$, $1-\alpha$, $1-\alpha^2$, $1-\alpha^3$ and $1-\alpha^5$
are needed to generate a group of units of full rank, but we then get 18 independent multiplicative relations among $\alpha$ and the $1-\alpha^j$. Now we look
at other $j$ for which the norm of $1-\alpha^j$ is no longer 1 but still factors into small prime factors which have already occurred for previous $j$,
and then try to form combinations which are units.  For instance, the norms of $\alpha^j-1$ are positive powers of 3 for $j=4$, 8, 12, 16, 20, 24,
36, and 40, and eliminating $\alpha^4-1$,
which together with the units generates the group of all elements of $F$ whose norm is a power of 3, we get 7 further relations. Proceeding in this way (in practice
it is more efficient to use the cyclotomic polynomials $\Phi_j(\alpha)$ rather than the numbers $\alpha^j-1$) gives
the 71 relations mentioned. By the time we get to $j=1000$, the norms are so huge that it seems clear that $\alpha^j-1$ will never again be a combination of
smaller values, and this could be proved by a finite effort if really required.

It seems very likely that this particular $\alpha$ gives the maximum of
$\dsize\frac{d(\alpha)}{[\Q(\alpha):\Q]}$ (indeed, quite possibly even of $d(\alpha)$) for all algebraic numbers, in which case there are probably no
valid ladders at all of order bigger than 16.  In any case, the study of the number $d(\alpha)$, motivated by the ladder concept, seems to be a very
interesting problem in the field of diophantine approximation.


\subheading{5. Existence of relations among polylogarithm values of arbitrarily high order} Let $S$ be a set of $s$ numbers in $\Q$ and
$X(S)$ the set of all $x\in\Q$ such that both $x$ and $1-x$ belong to the multiplicative group generated by the elements of $S$. The 
number of independent requirements on a combination $\xi=\sum_{x\in X}n_x[x]$ to belong to $\a m(\Q)$ is
$$(m-1)\binom{s+m-2}m+\sum\Sb 0\le r<m-2\\m-r\text{ odd}\endSb\binom{s+r-1}r$$ (the first term is the dimension of
$\sy {m-1}(V)\otimes V/\sy m(V)$ where $V=\langle S\rangle$ is $s$-dimensional; the other terms give the number of successive
conditions for the images of $\xi$ under the various maps $\iota_{\phi_{i_1}}\ldots\iota_{\phi_{i_r}}$ to map to 0 in $\r{m-r}\cong\Z$), which 
is a polynomial in $s$. Therefore if there are sets $S$ of arbitrarily large cardinality $s$ for which $|X(S)|$ grows more than
polynomially with $s$, then it follows that $\a m(\Q)$ has infinite rank and hence that there are infinitely many relations among the 
values of $L_m(x)$ ($x\in\Q$) for every $m$. The existence of such $S$ is the content of a theorem of Erd\"os-Stewart-Tijdeman [EST].
A simplified presentation of their proof was given in [Z] and a further simplification (with a slightly weaker bound, but not appealing
to any results from analytic number theory) is given here.
\proclaim{Theorem (Erd\"os-Stewart-Tijdeman)} For any $\varepsilon>0$ there exist sets $S$ of arbitrarily large cardinality $s$ for which
$\dsize|X(S)|>e^{(2-\varepsilon)\sqrt{s/\log s}}\,.$\endproclaim
\demo{Proof} Let $t$ and $u$ be large numbers, to be chosen presently, and let $A$ denote the set of products $p_1\ldots p_r$ with $r\le u$ 
and each $p_i$ a prime $\le t$. The cardinality of $A$ is then $\dsize\binom{u+\pi(t)}u$ which for $\pi(t)$ (number of primes $\le t$) much larger than $u$
equals $\dsize\frac{\pi(t)^{u+\text o(u)}}{u!}=\bigl(\frac{e\pi(t)}u\bigr)^{u+\text o(u)}$. The number of pairs $(a,c)$ with $a,\,c\in A$, $c>a$, is 
therefore equal to $\dsize\binom{|A|}2=\bigl(\frac{e\pi(t)}u\bigr)^{2u+\text o(u)}$. For each such pair the difference $b=c-a$ is an integer
between 0 and $t^u$, so by the pigeonhole principle there is a number $b$ which is expressible as $c-a$ in at least
$N=\dsize\binom{|A|}2/t^u=\bigl(\frac{e\pi(t)}{u\sqrt t}\bigr)^{2u+\text o(u)}$ ways. We maximize this by choosing $u=\pi(t)/\sqrt t$, giving
$\dsize N=e^{(2+\text o(1))\pi(t)/\sqrt t}$. 
Now take $S=\{b\}\cup\{p\le t,\text{ $p$ prime}\}$, with cardinality $s=1+\pi(t)=(1+\text o(1))\dsize\frac t{\log t}$; then each representation 
$b=c-a$ gives a distinct element $x=\dsize\frac bc$ of $X(S)$, so $|X(S)|>N=\dsize e^{(2+\text o(1))\sqrt{s/\log s}}\,.$\enddemo
\proclaim{Corollary} (1) $\a m(\Q)$ has infinite rank for all $m$. 
(2) For any $m$, there are infinitely many linearly independent relations over $\Q$ among the values of $L_m(x)$, $x\in\Q$.\endproclaim
\noindent{\bf Remarks. 1.} The original result of Erd\"os-Stewart-Tijdeman was slightly stronger in that 2 was replaced by 4 in the exponent
and $S$ was required to consist only of primes; for the proof of the stronger statement, see [EST] or [Z].

\noindent{\bf 2.} In part (1) of the corollary, we could have written ``$\a m(F)$ for any number field $F$" instead of $\a m(\Q)$, since
$\a m(F)$ contains $\a m(\Q)$, but of course the interesting question is whether one gets infinitely many {\it new} relations on
passing from $\Q$ to $F$, and this is not clear without doing more work.

\noindent{\bf 3.} Of course, part (2) of the corollary is not a corollary of the Erd\"os-Stewart-Tijdeman theorem alone, but of this theorem together with
Beilinson's deep result on the relation between the polylogarithm and regulator mappings.


\subheading{6. A conjecture on linear independence} We conjecture that the only linear relations over $\Q$ among polylogarithm values at
algebraic arguments are those which follow from the theory explained in \S1. More precisely, this says that if there is a relation
$$\sum_in_iP_m(x_i)=0\qquad(n_i\in\Z,\;x_i\in\qb)\,,\tag 1$$ (respectively $\sum n_iL_m(x_i)=0$ with $n_i\in\Z$ and $x_i\in\qb\cap\R$),
then \item{ }(i) $\sum n_ix_i$ satisfies the algebraic relation $(*_m)$, and \item{ }(ii) the conjugate equations $\sum n_iP_m(x_i^\sigma)=0$
($\sigma\in$Gal$(\qb/\Q)$), as well as the associated equations $\sum n_i\phi_1(x_i)\cdots \phi_r(x_i)P_{m-r}(x_i)=0$ and {\it their} conjugates for all 
$r\le m-2$ and all homomorphisms $\phi_1,\ldots,\phi_r:\qb^\times\to\Q$, also hold.

\noindent Since we further conjecture that the kernel of the maps $\vec P_m:\a m(F)\to\R^{n_\mp}$ is the group $\c m(F)$ defined by specializing functional
equations, we can state the combined conjecture more concisely by saying that the only relations (1) are specializations of functional
equations $\sum n_iP_m(\phi_i(t))=0$ ($\phi(t)\in\Q(t)$) to arguments $t\in\qb$. (This includes conditions (i) and (ii) since the arguments
of functional equations of $P_m$ automatically satisfy $(*_m)$ and since replacing $t\in\qb$ by $t^\sigma$ replaces each value $x_i=\phi_i(t)$
by $x_i^\sigma$.) This conjecture, which is discussed in more detail in \S10 of [Z], contains as a special case Milnor's conjecture [M] that the
only linear relations over $\Q$ of the Clausen function $\text{Cl}_2(\theta)=\sum_{n=1}^\infty\sin(n\theta)/n^2$ at arguments
$\theta\in\Q\pi$ are those arising from the distribution relations 
$$\sum_{m=1}^{|N|}\text{Cl}_2\bigl(\theta+\frac{2m\pi}N\bigr)=\frac1N\,\text{Cl}_2(N\theta)\,\qquad(0\ne N\in\Z).$$
The only evidence in its support is its naturalness and internal consistency and the fact that the many known examples of algebraic
relations among polylogarithms all conform to it. On the theoretical level nothing is known; so far as I know, for instance, one cannot prove
that there is even a single pair of values $D(x)$, $D(y)$  with $x$ and $y$ algebraic which are linearly independent over $\Q$, or even that
there is a single value of $D(x)$ which is not a rational number! The conjecture is also particularly daring because we know from \S5 that there are linear
dependences among values of the $m$th polylogarithm function for arbitrarily large values of $m$, but for $m>6$ do not know whether these functions have
any non-trivial functional equations at all.


\subheading{7. Functional equations} The basic fact here, already mentioned in \S1, is that the only requirement for a functional
equation is the algebraic condition $(*_m)$, i.e., that for any combination $\sum n_i[x_i]$ of functions (of one or several variables $t$) $x_i(t)$
which satisfies $(*_m)$ the corresponding sum of polylogarithms $\sum n_iP_m(x_i(t))$ is independent of $t$. In this section we discuss how this criterion
can be used to check known functional equations and to find new ones in an algorithmic way. We will express all functional equations in terms of $P_m$
rather than $\l m$ in order to have well-defined values for complex arguments and to avoid lower-order terms in the equations; for real values of the
arguments, the function $L_m$ would do just as well because of the relation between $L_m$ and $P_m$ mentioned at the end of $\S1$. A striking property
of many of the functional equations is their high level of symmetry; we will emphasize this aspect in our discussion. 

\subheading{Example 1. The 5-term relation for the dilogarithm} The basic functional equation of the dilogarithm function is the 5-term relation
(cf\. Chapter I, section 1.2), which in our notation says that $D\bigl(R_5(x,y)\bigr)=0$ for any $x$ and $y$, where $R_5(x,y)$ denotes the formal linear combination
$$R_5(x,y)=\bigl[x\bigr]+\bigl[y\bigr]+\bigl[\frac{1-x}{1-xy}\bigr]+\bigl[1-xy\bigr]+\bigl[\frac{1-y}{1-xy}\bigr]\tag1$$
of elements in $\Q(x,y)$. (The other forms given by Spence, Hill, Abel and Kummer are equivalent to this by the 1-variable relations of
Chapter I, 1.1.) To prove it in our language, we have to check that $R_5(x,y)$ is in the kernel of the map $\beta_2:[z]\mapsto[z]\w[1-z]$. We calculate
$$\align\beta_2\bigl(R_5(x,y)\bigr)&=[x]\w[1-x]+[y]\w[1-y]+\bigl([1-x]-[1-xy]\bigr)\w\bigl([x]+[1-y]-[1-xy]\bigr)\\
&+[1-xy]\w\bigl([x]+[y]\bigr)+\bigl([1-y]-[1-xy]\bigr)\w\bigl([y]+[1-x]-[1-xy]\bigr)=0.\endalign$$
This is the simplest example of the use of the calculus with wedge products in checking functional equations (and also, as far as the dilogarithm is
concerned, the basic example, since it is apparently the case that all functional equations of the dilogarithm are consequences of the 5-term relation.)

We use this simple example to give a first illustration of the comments on symmetry made at the beginning 
of the section. First of all, we can check that $R_5$ has a cyclic symmetry, i.e., 
$$R_5\bigl(x,\,y\bigr)=R_5\bigl(y,\,\frac{1-x}{1-xy}\bigr)=R_5\bigl(\frac{1-x}{1-xy},\,1-xy\bigr)
 =R_5\bigl(1-xy,\,\frac{1-y}{1-xy}\bigr)=R_5\bigl(\frac{1-y}{1-xy},\,x\bigr).$$
However, this is only part of the full symmetry group. The basic invariance property of the dilogarithm is that
for any $x$ the six numbers $D(x)$, $D(1/x)$, $D(1-1/x)$, $D(x/(x-1))$, $D(1/(1-x))$ and $D(1-x)$ are equal up to sign. This 6-fold symmetry plays a 
role so often in the following that we introduce the special notation 
$$x'\six x\;\Leftrightarrow\;x'\in\{x,\,\frac1x,\,1-\frac1x,\,\frac x{x-1},\,\frac1{1-x},\,1-x\}\,.$$
 A symmetric way to express the $\six$-invariance is to say that the function $D([a,b,c,d])$, where $[a,b,c,d]=\dsize\frac{a-c}{a-d}\frac{b-d}{b-c}$ 
denotes the cross-ratio of $a$, $b$, $c$ and $d$, is up to sign a symmetric function of its four
arguments, since changing the order of four numbers replaces their cross-ratio $x$ by a number $x'\six x$. More precisely, 
$$D\bigl([x_{\pi(1)},\,x_{\pi(2)},\,x_{\pi(3)},\,x_{\pi(4)}]\bigr)=\sgn(\pi)\,D\bigl([x_1,x_2,x_3,x_4]\bigr)\tag2$$
for any $x_i\in\P$ ($1\le i\le4$) and any $\pi\in\frak S_4$, the symmetric group on 4 letters. To verify this in our terminology we compute
$$\align\beta_2\bigl([x_1,x_2,x_3,x_4]\bigr)
&=\biggl[\frac{(x_1-x_3)(x_2-x_4)}{(x_1-x_4)(x_2-x_3)}\biggr]\,\w\,\biggl[\frac{(x_1-x_2)(x_3-x_4)}{(x_1-x_4)(x_2-x_3)}\biggr]\\
&=\frac12\,\sum_{\pi\in\frak S_4}\sgn(\pi)\,[x_{\pi(1)}-x_{\pi(2)}]\w[x_{\pi(2)}-x_{\pi(3)}]\endalign$$
(this has only 12 terms rather than 24, because the $\pi$th summand is invariant under $\pi\mapsto\pi\tau$ where $\tau$ is $1\leftrightarrow3$). This
has the desired invariance property under the action of $\frak S_4$, so $D([x_1,x_2,x_3,x_4])$ also does. Now the 5-term relation becomes simply
$$\frac1{24}\,\sum_{\pi\in\frak S_5}\sgn(\pi)\,D\bigl([x_{\pi(1)},\,x_{\pi(2)},\,x_{\pi(3)},\,x_{\pi(4)}]\bigr)=0,\qquad(x_1,\ldots,x_5\in\P)\tag3$$
(this has only 5 terms rather than 120 because the $\pi$th summand is unchanged by $\pi\mapsto\pi\tau$, $\tau\in\frak S_4$), with a symmetry group of order 120.


\subheading{Example 2. The 9-term relation for the dilogarithm} As a second example, we consider the 3-variable functional equation (2.40) of Chapter I, 2.3.2.  
This equation has an obvious 8-fold symmetry generated by the involutions $x\leftrightarrow y$, $v\leftrightarrow w$ and $(x,y)\leftrightarrow (v,w)$, but
in fact has a symmetry group of order 72, a typical non-obvious symmetry being given by $(x,y,v,w)\mapsto(y,xy/vw,y/w,y/v)$. To make the symmetries obvious, 
we first define a $3\times3$ matrix $$Z=\bigl(z_{ij}\bigr)_{i,j=1,2,3}=\pmatrix x&1/v&v/x\\1/w&y&w/y\\w/x&v/y&xy/vw\endpmatrix\,.\tag4$$
Then the functional equation becomes simply $\sum_{i,j}D(z_{ij})=0$. The constraints on the $z_{ij}$ are
$$\prod\limits_jz_{ij}=1\;\quad(\forall i)\,,\qquad\qquad\prod\limits_iz_{ij}=1\;\quad(\forall j)\tag 5$$ 
and (taking the indices $i$ and $j$ modulo 3)
$$(1-z_{i,j})(1-z_{i+1,j+1})=(1-z_{i,j+1}^{-1})(1-z_{i+1,j}^{-1})\qquad(\forall i,\,j)\,.\tag 6$$
Indeed, (5) is equivalent to the fact that $Z$ can be expressed by (4) for {\it some} numbers $x$, $y$, $v$ and $w$, and (6) for $i=j=1$
is just the constraint $(1-x)(1-y)=(1-v)(1-w)$. A calculation shows that this equation implies the validity of (6) for the other 8 values of the
indices $(i,j)$. Thus we have the symmetry group of order 72 given by permuting the indices $i$, permuting the indices $j$, and interchanging the roles
of $i$ and $j$. However, this is still not satisfactory because we cannot yet ``see" why equation (6) for one pair of indices $(i,j)$ implies (in conjunction
with (5)) its truth for all $i$, $j$. To remedy this, we choose new coordinates $t_{ij}$ ($i,\,j\in\Z/3\Z$) and set
 $z_{ij}=t_{i,j}\,t_{i,j+1}^{-1}\,t_{i+1,j}^{-1}\,t_{i+1,j+1}\,$. Then the equations (5) are true identically and each of the 9 equations (6) is equivalent to
the condition $\det(T)=0$, where $T=(t_{ij})$. (The set of $3\times3$ matrices $T$ with $\det(T)=0$ is 8-dimensional, but there are only 3 free parameters because
replacing $t_{ij}$ by $\lambda_i\mu_jt_{ij}$ for any $\lambda_i$ and $\mu_j$ leaves $z_{ij}$ unchanged.) Finally, we give a parametric solution of $\det(T)=0$ by
setting $t_{ij}=u_i-v_j$. Then the constraints (5) and (6) are automatically satisfied and $z_{ij}$ is simply the cross-ratio of the four numbers 
$u_i$, $u_{i+1}$, $v_j$ and $v_{j+1}$. Therefore the functional equation can be written
$$\sum_{i,\,j}D\bigl([u_i,\,u_{i+1},\,v_j,\,v_{j+1}]\bigr)=0\qquad\text{for any 6 points $u_i$, $v_j\in\P$\;($i,j\in\Z/3\Z$)}.$$
Now not only the 72-fold symmetry of the equation is obvious (permute the $u_i$'s or the $v_j$'s or interchange $\bold u$ and $\bold v$), but also its proof:
applying (3) to the 5-tuple $(u_i,u_{i+1},v_1,v_2,v_3)$ gives
$$\sum_jD\bigl([u_i,\,u_{i+1},\,v_j,\,v_{j+1}]\bigr)=D\bigl([u_i,v_1,v_2,v_3]\bigr)-D\bigl([u_{i+1},v_1,v_2,v_3]\bigr)$$
for each $i$, and the sum of this over $i\pmod3$ vanishes. Of course, we can now go back and give a non-motivated and non-symmetric proof of the equation
in its original form by adding the three 5-term relations $R_5\bigl(x,\,1/v\bigr)$, $R_5\bigl(1/w,\,y\bigr)$ and $R_5\bigl(w/x,\,v/y\bigr)$ and
using the 1-variable functional equations of the dilogarithm to simplify the result.

\subheading{Example 3. The 9-term relation for the trilogarithm} As the first example of the verification of a functional equation for a higher-order
polylogarithm we consider Kummer's 2-variable equation for the trilogarithm. Again we use $P_3$ rather than $\l3$ in order to eliminate lower-order terms
(the function $L_3$ would do just as well if we restricted our attention to real values of the variables). The functional equation in question 
(cf\. Chapter 2, 3.2.2.3) has the form $P_3\bigl(R_9(x,y)\bigr)=2\zeta(3)$, where $R_9(x,y)$ denotes the element
$$\align R_9(x,y)=&2\bigl[x]+2\bigl[y]+2\bigl[\frac{x(1-y)}{x-1}]+2\bigl[\frac{y(1-x)}{y-1}]+2\bigl[\frac{1-x}{1-y}]+2\bigl[\frac{x(1-y)}{y(1-x)}\bigr]\\
&-\bigl[xy]-\bigl[\frac xy]-\bigl[\frac{x(1-y)^2}{y(1-x)^2}]\tag7\endalign$$ of $\Q(x,y)$.
This equation has an obvious symmetry group of order 8 (invert $x$ or $y$ or interchange $x$ and $y$), but in fact has a symmetry group of order 24. We can
write it symmetrically in two ways. 

{\it Symmetric form: first version.} Let $H$ be the hypersurface in $\Bbb P^3(\C)$ consisting of 4-tuples $(a_1:\,a_2:\,a_3:\,a_4)$ with $\sum_i a_i=0$, with 
the obvious action of the group $\frak S_4$. Let $V$ be the 3-dimensional space generated multiplicatively by all ratios $a_i/a_j$ 
and $W\supset V$ the 6-dimensional space generated by all quotients $(a_i+a_j)/a_i$, $j\ne i$ (this space has dimension only 6 because up to sign there are only
three distinct quantities $a_i+a_j$). Each of the elements $x=-a_i/a_j$ ($j\ne i$) and $a_ia_j/a_ka_l$ ($i$, $j$, $k$, $l$ distinct) has the property that $x\in V$
and $1-x\in W$, and up to inversion there are exactly 9 of them.
The argument $R_9$ of Kummer's functional equation can then be written symmetrically as
$$2\,\bigl(\,\bigl[-\frac{a_1}{a_2}\bigr]\,+\,\text{5 permutations}\,\bigr)-\bigl(\,\bigl[\frac{a_1a_2}{a_3a_4}\bigr]\,+\,\text{2 permutations}\,\bigr)\,.$$
(To see this, normalize the common value of $a_1+a_2$ and $-a_3-a_4$ to be 1, in which case $H$ is parametrized by two parameters
$x=-\frac{a_1}{a_2}$, $y=-\frac{a_3}{a_4}$ as $\bigl\{\bigl(\frac x{x-1}:\frac1{1-x}:\frac y{1-y}:\frac1{y-1}\bigr)\bigr\}$.) The proof of $P_3(R_9)=$ constant (the value of the
constant is then found by specializing to $x=0$, $y=1$) can be obtained easily by computing $\beta_3(-a_1/a_2)$ and $\beta_3(a_1a_2/a_3a_4)$ and symmetrizing with respect to
the group action.


{\it Symmetric form: second version.} As with the dilogarithm, we interpret the arguments of the trilogarithms as cross-ratios of 4-tuples of points in $\P$.
However, we no longer have the sixfold symmetry of the function $D(x)$, but only the twofold symmetry $P_3(x)=P_3(1/x)$. This means that the argument of $P_3(x)$
must be interpreted as a cross-ratio $[a,\,b,\,c,\,d]$ of four points  $a,\,b,\,c,\,d\in\P$ where the only symmetries allowed are the ones generated
by the interchanges of $a$ and $b$, of $c$ and $d$, or of $(a,b)$ and $(c,d)$, i.e. $P_3([a,b,c,d])$ depends only on the unordered pair of unordered pairs $\{\{a,b\},\,\{c,d\}\}$. 
To emphasize this, we write the cross-ratio $x$ as $[a,b;c,d]$ rather than $[a,b,c,d]$, although the definition is the same as before.
Now suppose that we have six points $p_\nu\in\,\P$ and that there is an involution $\tau$ on $\P$ which interchanges these points in pairs.
We consider all cross-ratios $[a,b;c,d]$ of four distinct points $a$, $b$, $c$, $d$ from the set $\{p_\nu\}$ for which 
$\{a,b\}$ is disjoint from $\{\tau(c),\tau(d)\}$. Using the symmetries of $[a,b;c,d]$ and the invariance of the cross-ratio under automorphisms
of $\P$, we find that there are 9 of these, 3 of the form $[a,\tau(a);c,\tau(c)]$ and 6 of the form $[a,\tau(a);c,d]$
with $d\ne\tau(c)$. Then $R_9$ is just twice the sum of the latter 6 minus the sum of the former 3. 
(Take the $p_\nu$ to be 0, $\infty$, 1, $xy$, $x$ and $y$ with the involution $\tau:t\mapsto xy/t$.) The equivalence of this form of the functional
equation to the one just given is seen by taking for $\{p_\nu\}$ the 6 numbers $a_i+a_j$ ($1\le i<j\le 4$) and for $\tau$ the involution $t\mapsto-t$.
The symmetry group appears now not as $\frak S_4$, but as the semi-direct product of $\frak S_3$ (permute the three orbits of $\tau$ on $\{p_\nu\}$) with a Klein 4-group
(interchanging the two elements within each orbit is an automorphism of order 2, but the product of these three automorphisms is just $\tau$ and
has no effect on the cross-ratio).

The functional equations discussed up to now are classical, but the method of verifying them
can also be used to discover new equations. The basic desideratum, as in the case of relations among special values, is to find as many $x$ as possible such that
all the $x$ and $1-x$ belong to subspaces $V$ and $W$ of $F^\times$ of small dimension. Typically one first chooses the generators of $V$ (certain irreducible
polynomials in one or several variables), preferably with a lot of symmetry to reduce the number of independent conditions which have to be checked later, and
then looks for many $x$ in $V$ for which the prime factors of the elements $1-x$ are repeated many times; then one computes $\beta_m(x)$ for these $x$'s and some
small $m$ and uses linear algebra to solve if possible the system of equations which expresses that a linear combination of them vanishes. This algorithm, which can
be carried out by hand in simple cases, can also be programmed, although not easily. This has been carried out by H\. Gangl, who in this way found a large number
of new functional equations. In the remainder of the appendix we present a few of these.
\subheading{Example 4. Functional equations for the tetralogarithm}  The simplest of Gangl's functional equations is the 9-term equation
$$\aligned&2\,\bigl[\te{z(1-z)}+\te{-\frac z{(z-1)^2}}+\te{\frac{z-1}{z^2}}\bigr]\\
-&3\,\bigl[\te{\frac1{1-z+z^2}}+\te{\frac{(1-z)^2}{1-z+z^2}}+\te{\frac{z^2}{1-z+z^2}}\bigr]\\
-&6\,\bigl[\te{\frac{1-z+z^2}{z(z-1)}}+\te{\frac{1-z+z^2}z}+\te{\frac{1-z+z^2}{1-z}}\bigr]\,=\,0\,.\endaligned\tag8$$ This
has a 6-fold symmetry $G_9(z')=G_9(z)$ for $z'\six z$ and up to this symmetry has only three rather than nine terms, as shown
by the square brackets.  

More interesting is the equation                  \def\te#1{P_4\bigl(#1\bigr)}
$$\align&\;\bigl[\te{-\frac{bd}a}+\te{-\frac{ac}d}+\te{-\frac{ab^3}{c^2d}}
  +\te{-\frac{c^3d}{ab^2}}+\te{-\frac d{ab^2}}+\te{-\frac a{c^2d}}\bigr]\\
&+\bigl[\te{-\frac{ad^2}b}+\te{-\frac{a^2d}c}\bigr]+2\bigl[\te{\frac b{c^2}}+\te{\frac c{b^2}}+\te{bc}\bigr]\\
&+3\bigl[\te{-\frac b{ad}}+\te{-\frac c{ad}}+\te{-\frac1{ad}}\bigr]\\ 
&+3\bigl[\te{-cd}+\te{-ab}+\te{-\frac dc}+\te{-\frac ab}+\te{-\frac{ab}{c^2}}+\te{-\frac{cd}{b^2}}\bigr]\\ 
&+6\,\bigl[\te{\frac ca}+\te{\frac bd}+\te{\frac1d}+\te{\frac1a}+\te{\frac b{cd}}+\te{\frac c{ab}}\bigr]\;=\;0\,, \endalign$$ 
in two variables $y$ and $z$, where we have abbreviated $$a=1-z+yz,\quad b=-y,\quad c=y-1,\quad d=1-y+yz$$
This equation has an obvious symmetry under the involution $P:(a,b,c,d)\mapsto(d,c,b,a)$,
corresponding to $(y,\,z)\mapsto(1-y,\,1-z)$, and a less obvious one under the involution $Q:(a,b,c,d)\mapsto(a,b^{-1},cb^{-1},db^{-1})$,
corresponding to $(y,z)\mapsto(y^{-1},\,-yz)$. These two involutions generate a group of order 12 under which the 26 terms of the functional
equations fall into only 6 orbits (grouped by square brackets in the formula above). 
One can introduce new variables which make the symmetries obvious.  Let $F$ be the function field
(over $\Q$) of the variety $$X\;=\;\bigl\{(t_1:t_2:t_3:t_4:t_5)\in\Bbb
P^4\;\mid\;\tsize\sum\limits_{i=1}^5t_i=0,\quad\sum\limits_{i=1}^3t_i^{-1}=0\,\bigl\}\,,$$  with an obvious symmetry of the group $G=\frak
S_3\times\frak S_2\subset\frak S_5$. We can parametrize $X$ by $$\bigl(t_1:t_2:t_3:t_4:t_5\bigr)=\bigl(-y:y-1:y(1-y):y-y(1-y)z:(y-1)^2+y(1-y)z\,\bigr)\,,$$
(set $y=t_3/t_2$, $z=-(t_1+t_4)/t_3$) and under this identification $G$ is identified with the symmetry group $\langle P,\,Q\rangle$ of the functional equation.
The 26 arguments $x$ of $P_4$ satisfy $x\in V$, $1-x\in W$ where $V$ is the 4-dimensional subspace of $F^\times\otimes_{\Z}\Q$ spanned by
the quotients $t_i/t_j$ ($i\ne j$) and $W$ the 11-dimensional space spanned by $V$ and the elements $(t_i+t_j)/t_i$ ($i\ne j$, $\max\{i,j\}\ge4$). Using this
description and the symmetry one can easily check that the linear combination of the $x$'s specified by the functional equation is in the kernel of $\beta_4$.

\subheading{Example 5. A 2-variable functional equation for the hexalogarithm} Gangl found one-variable functional equations for the penta- and hexalogarithm
which had a similar structure to equation (8) above, namely the arguments of the polylogarithms involve only the
irreducible polynomials $z$, $1-z$ and $1-z+z^2$ and the whole equation is unchanged if $z$ is replaced by any $z'\six z$. The functional equation for the
hexalogarithm, in particular, has 60 terms forming 13 orbits under the 6-fold symmetry and can be written efficiently as
\def\i#1#2#3#4{T^{#1}_{\{#2,#3,#4\}}} $$\align&\quad3\i-015+4\i+{-5}03+5\i-{-4}33-20\i-013-20\i-004-60\i-{-3}12+90\i-{-1}12\\
&+95\i-{-1}03+120\i+{-2}11-180\i+{-1}12-360\i+011+540\i-011-544\i+002=0\,,\endalign$$  where for a set of three integers $\alpha$, $\beta$, $\gamma$ with even sum we have set
 $$\i\pm\alpha\beta\gamma=\sum_{\{a,b,c\}=\{\alpha,\beta,\gamma\}}P_6\biggl(\pm\frac{(-1)^az^b(1-z)^c}{(1-z+z^2)^{(a+b+c)/2}}\biggr)$$ 
(the sum is over all permutations and has 6 terms in general but only 3 if two of $\alpha$, $\beta$ and $\gamma$ coincide). Later he found a second functional
equation with 89 terms and where the prime factors of the hexalogarithm arguments involved four prime factors $z$, $1-z$, $1+z$ and $1-z+z^2$. 
Many of the coefficients in these two equations agreed, suggesting that they might be different specializations of the same two-variable equation. After a
considerable amount of trial and error, guided by the symmetry, it was found that this is true and that the two-variable functional equation has an 18-fold symmetry:
\proclaim{Theorem} Let $y$ and $z$ be variables and denote by $y_i$ ($1\le i\le6$) and $w_j$ ($1\le j\le3$) the elements $y'\six y$ and $z'(1-z')$, $z'\six z$.
Then $$\sum_{a,b}\,n_{a,b}\,\frac1{\mu_{a,b}}\,\sum_{i=1}^6\sum_{j=1}^3P_6\biggl(\bigl(\frac{w_j}{y_i}\bigr)^a\bigl(\frac{1-w_j}{1-y_i}\bigr)^b\biggr)\tag9$$
is independent of $y$, where the coefficients $n_{a,b}$ and $\mu_{a,b}$ are given by the following table:
$$\vbox{\offinterlineskip\halign{$\hfil#\hfil$&\ \vrule#\ &&$\hfil\phantom{x}#\phantom{x}\hfil$\cr  
(a,b)&&(-2,\,3)&(-1,\,-1)&(-2,\,1)&(1,\,-2)&(1,\,0)&(0,\,1)&(1,\,-1)\cr  \omit &height5pt&\omit&\cr
\noalign{\hrule}\omit &height5pt&\omit&\cr n_{a,b}&&3&4&5&20&60&90&180\cr \mu_{a,b}&&3&2&2&2&1&1&1\cr}}$$\endproclaim
\noindent{\bf Remarks. 1.} The value of (9) can be found by specializing $y$ in any way we want. Taking $y=\infty$ (or 0 or 1, which are equivalent under $\six$),
we find that it equals $60P_6(G_9(z))$, where $G_9(z)$ is the combination of arguments occurring in Gangl's tetralogarithmic functional equation (8).
\newline\noindent{\bf 2.}The coefficient of each inner double sum in (9)
has been written as $n_{a,b}/\mu_{a,b}$ rather than simply $n_{a,b}$ because the 18 terms of the double sums occur with multiplicity $\mu_{a,b}$, so that
the $(a,b)$th summand in (9) in fact consists of $18/\mu_{a,b}$ terms with coefficient $n_{a,b}$. Thus there are 87 terms altogether, forming 7 orbits under
a group of order 18, or 96 terms forming 8 orbits if we include the ``constant term" as given in the previous remark.
\newline\noindent{\bf 3.} Gangl's original 1-variable equations are obtained by specializing to $y=z$ and $y=-z$.
\newline\noindent{\bf 4.} The coefficients $n_{a,b}$ are determined by the requirement that the homogeneous polynomial $\sum n_{a,b}(aX+bY)^5$ vanish identically
(since this polynomial is a combination of 6 monomials $X^kY^{5-k}$ and there are 7 indices $(a,b)$, this has a solution). To see the necessity
of this condition, note that the arguments
$x$ of the hexalogarithms in (9) belong to the 16-dimensional space generated by the irreducible polynomials $y_i/w_j$ and $(1-y_i)/(1-w_j)$ 
($1\le i\le6$, $1\le j\le3$), while $1-x$ belongs to the 34-dimensional space with the additional generators $p_{i,j}=y_i-w_j$. 
The contribution of one of the new primes $p=p_{i,j}$ to $\beta_6(x)$, where $x$ is one of the arguments in (9), is 
$\text{ord}_p(x)\,[x]^5\otimes[p]\in\sy5(F^\times)\otimes F^\times$ ($F=\Q(y,z)$), so a short consideration shows that the total coefficient
of $[p_{i,j}]$ under $\beta_6$ equals $\sum n_{a,b}(aX+bY)^5$ with $X=[y_i/w_j],\;Y=[(1-y_i)/(1-w_j)]\in F^\times$. Thus the functional equation stated in the theorem
can only hold for the values of $n_{a,b}$ given in the table; that it actually does hold depends on luck (at least at our present stage of understanding), since
the coefficients are already uniquely determined by requiring that the ``new" primes $p_{i,j}$ drop out under the map $\beta_6$, and we have no further freedom
to ensure that the ``old" primes also give a zero contribution modulo $\sy6(F^\times)$.

\heading References\endheading
\ref\no[AL]\by M\. Abouzahra and L\. Lewin, The polylogarithm in algebraic number fields\jour J\. Number Theory\vol21\yr1985\pages214--244\endref
\ref\no[B]\by A\. Borel, Cohomologie de $SL_n$ et valeurs de fonctions z\^eta aux points entiers\jour Ann\. Sc\. Norm\. Sup\. Pisa\vol4\yr1977\pages613--636\endref
\ref\no[BD]\by A.A\. Beilinson and P\. Deligne, in preparation\endref
\ref\no[EST]\by P\. Erd\"os, C\. Stewart and R\. Tijdeman, Some diophantine equations with many solutions \jour Comp\. Math\.\vol 66 \yr 1988 \pages 36--56\endref
\ref\no[G]\by A\.B\. Goncharov, Geometry of configurations, polylogarithms and motivic cohomology, preprint, 1990 \endref
\ref\no[L]\by L\. Lewin, {\it Polylogarithms and Associated Functions}\publaddr North-Holland, New York, 1981\endref
\ref\no[M]\by J\. Milnor, Hyperbolic geometry: the first 150 years\jour Bull AMS\vol 6\yr1982\pages9--24\endref
\ref\no[S]\by A.A\. Suslin, Algebraic $K$-theory of fields, in {\it Proceedings of the International Congress of Mathematicians 1986}
  \publaddr AMS, 1987, pp\. 222--244\endref
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\newpage {\bf Remark.} For computing numerical examples in number fields it is useful to have a simple and rapidly convergent formula for
calculating $Li_m(z)$. For $z$ small (say $|z|<1/2$), one can of course simply use the defining series of the polylogarithm, and for $z$ large
the functional equation relating the polylogarithms of $z$ and $1/z$. For $z$ near the unit circle, a convenient (and pretty) formula which was
noticed by Henri Cohen and myself is $$Li_m(e^x)=\sum_{n=0}^\infty \,\zeta(m-n)\,\frac{x^n}{n!}\,,$$ where the meaningless term $\zeta(-1)$ is
to be replaced by $1+\frac12+\cdots+\frac1{m-1}-\log(-x)$. This formula, easily proved by $m$-fold differentiation, works for all $x$ of absolute value
less than $2\pi$ and hence gives $Li_m(z)$ for all $z$ with $.005<|z|<230$.

\end





 $$\sum_{j=1}^3\biggl(60P_6\bigl(w_j\bigr)+90P_6(1-w_j)+180P_6\bigl(\frac{w_j}{w_j-1}\bigr)\biggr)\,.$$ 

We will define $\a m(F)\subseteq\g m(F)\subset\f F$ inductively in such a way that the image of $\a m(F)$ is contained in a lattice $\r m=\r m(F)\subset\R^{n_\mp}$

can define $\a m(F)$ inductively in such a way that the image of, is contained in a lattice in $\R^{n_\mp}$. Assuming that we have done this for all $m'<m$, then $\a
m(F)$ is defined as the set of $\xi\in\g m(F)$ for which $\iota_\phi(\xi)$  belongs to $\text{Ker}\bigl(P_{m-1}:\a{m-1}(F)\to\R^{n_\pm}\bigr)\subseteq\a{m-1}(F)\subseteq\g{m-1}(F)$
for every homomorphism $\phi:F^\times\to\Z$.  In other words, each of the combinations $\iota_{\phi_1}\ldots\iota_{\phi_r}(\xi)\in\g{m-r}(F)$ for $o<r\le m-2$ should
belong to the inductively defined group $\a{m-r}(F)$ and be mapped under $P_{m-r}$ to the zero element of the appropriate lattice.
