Monodromy of Polylogarithms
Minh Hoang Ngoc
LIFL, Université de Lille I
Algorithms Seminar
July 6, 1998
[summary by David M. Bradley]
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Abstract
Generalized polylogarithms are complex, multivalued functions with
singularities at z=0 and z=1. We calculate the monodromy at the
two singularities. As opposed to the classical
polylogs [11, 12],
the monodromy
of generalized polylogs involves the socalled ``multiple zeta
values,'' [14]
which play an important role in number theory, knot
theory [4, 6, 5, 10],
and physics [7, 9].
Via monodromy of polylogs, Radford [13] showed
that the Calgebra of polylogs
is isomorphic to the Calgebra of noncommutative polynomials in two
variablesa ``shuffle algebra'' freely generated by the socalled
Lyndon words.
Here, monodromy is used to give an
induction proof of the linear independence of the polylogarithms.
We also obtain a Gröbner basis of the polynomial relations between
``multiple zeta values'' using the techniques of noncommutative algebra.
By expressing multiple zeta values in terms
of the Gröbner basis, one obtains symbolic algebraic proofs of
relations between multiple zeta values.
1 Polylogarithms and Combinatorics on Words
Let X={x_{0},x_{1}}. To any word w=x_{0}^{s}_{1}^{1}x_{1}x_{0}^{s}_{2}^{1}x_{1}
··· x_{0}^{s}_{k}^{1}x_{1}
we associate the multiindex s=(s_{1},s_{2},...,s_{k}) and define
the generalized polylogarithm
Li_{w}(z)=Li 
_{s}(z)= 

å 
n_{1}>n_{2}>...>n_{k}>0 



.

The associated multiple zeta value
is z_{w}=z(s)=Li_{w}(1)=Li_{s}(1).
The shuffle product is defined on words by
the recursion
xu W yv = x(u W yv) + y(xu W v),
where x,yÎ X and u and v are words on X.
We can extend the shuffle product linearly to the noncommutative
polynomials Qá Xñ. The resulting polynomial algebra, denoted
Sh_{Q}(X) is commutative and associative.
The Lyndon words L are those nonempty
words on X that are inferior to each of their right factors
in the lexicographical order. They are algebraically independent
and generate Sh_{Q}(X), thus forming a transcendence
basis. More precisely,
a theorem of Radford [13]
states that the algebra Sh_{Q}(X)
is isomorphic to the polynomial algebra generated by the Lyndon
words, i.e. Q[L].
2 Relations between Multiple Zeta Values
There are countless relations between multiple zeta
values [1, 3, 2].
We content
ourselves here with providing only two examples:
z(2,1)=z(3)
and z(2,2,1)= 

z(5)
+3z(2)z(3).

It turns out that a large class of relations can be
explained by the collision of two distinct
shuffles obeyed by the multiple zeta values. We've already seen
one type of shuffle. It provides relations
of the form z_{uW v}=z_{u}
z_{v}. A second type of shuffle provides relations of the form
z_{u*v}=z_{u}z_{v} and is defined by the recursion
(s_{1},s) * (t_{1},t) = (s_{1},s*(t_{1},t)) + (t_{1},(s_{1},s)*t)+(s_{1}+t_{1},s*t),
where we have used the multiindex notation s=(s_{2},s_{3},...,s_{k}),
t=(t_{2},t_{3},...,t_{r}) of Section 1.
With a slight abuse of notation, we define a map z: w® z_{w},
extended linearly in the natural way to Qá Xñ. Then z
is a Qalgebra homomorphism which respects both shuffle products.
Thus, if I is
the ideal generated by the words u W v  u*v, then I Í
kerz. We can compute a Gröbner basis for the ideal
I up to any given order using only symbolic computation. The first
relation above is the unique basis element of order 3. The second
relation above is one of five basis elements of order 5.
3 Monodromy of Polylogarithms
To compute the monodromy, we use the standard keyhole contours about
the two singularities z=0 and z=1. The monodromy is given by
where the remaining terms are linear combinations of polylogarithms coded
by words of lengths less than the length of w.
For example, using the computational package Axiom, we find that
M_{1}Li 

=Li 

,
M_{1}Li 

=Li 

2p i,
M_{1}Li 

=Li 

2p iLi 

,

and so on.
The generating series of the generalized polylogarithms is
with the convention that Li_{x}_{0}^{n}(z) = (log z)^{n}/n!.
Drinfel'd's differential equation [8, 9]
is satisfied, with boundary condition
L(e)=exp(x_{0}log e)+O((e)^{1/2}) as e®0+.
It turns out that L is a Lie exponential, and this fact can be used
to obtain asymptotic expansions of the generalized polylogarithms at
z=1.
4 Independence of Polylogarithms
Theorem 1
The functions Li_{w} with wÎ X^{*} are Clinearly
independent.
Corollary 1
The Calgebra generated by the Li_{w} is isomorphic
to Sh_{C}(X). By Radford's theorem, the generalized
polylogarithms coded by Lyndon words form an infinite transcendence
basis.
Corollary 2
Each generalized polylogarithm Li_{w} has a unique
representation as a Qpolynomial in polylogarithms coded by
Lyndon words. The classical [11, 12]
polylogarithms Li_{k},
which are coded by the Lyndon words x_{0}^{k1}x_{1}, are algebraically
independent.
Proof.[Proof of Theorem 1]
Given n³ 0, assume that


l_{w} Li_{w} = 0, l_{w}ÎC,
(1) 
where w denotes the length of the word w. We prove by induction
on n that l_{w}=0 for all w, the case n=0 being trivial.
Rewrite (1) as
Applying the operators (M_{0}Id) and (IdM_{1})
on this latter expression, yields two new linear relations
ì ï ï í ï ï î 
2p i 

l 

Li 
_{u}
+ 

µ_{u}Li_{u} 

=0, 
2p i 

l 

Li 
_{u}
+ 

n_{u}Li_{u} 

=0, 

for certain coefficients µ_{u} and n_{u}. By the induction hypothesis,
the coefficients l_{ux}_{0} and l_{ux}_{1} with u=n1
all vanish (as well as the coefficients µ_{u} and n_{u}).
Consequently,
whence l_{w}=0 for all w, again by the induction hypothesis.
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