p7

[anlytic]-->[analytic]

Juste avant 
"We will use the following increasing filtrations"

We recall that, for $a\in X$ and $w\in X^*$, the partial degree $|w|_a$ is the 
number of occurrences of $a$ in $w$, it is defined by the recursion 
$$
|1_{X^*}|_a=0\ ;\ |bu|_a=\delta{b,a}+|u|_a\ .
$$ 
Or course the lenght of the word is the sum of the partial degrees i.e. 
$|w|=\sum_{x\in X}|w|_x$. 
The function $a\mapsto |w|_a$ belongs to $\N^{(X)}$ (finitely supported functions from $X$ to $\N$). For $\al\in \N^{(X)}$, we note $\ncp{\C_{\leq \al}{X}$, the set of polynomials $Q\in \ncp{\C}{X}$ such that $supp(Q)\subset X^{\leq \al}$ i.e.
$$
\scal{Q}{w}\not=0\Longrightarrow (\forall x\in X)(|w|_x\leq \al(x))
$$
In the same way, we consider the filtration by total degree (length) 
$$
\ncp{\C_{\leq n}{X}=\sum_{|\al|\leq n} \ncp{\C_{\leq \al}{X}\ .
$$

p7

[between the Differential Galois group of a differential equation of type \mref{LSE} (acting on the right) ]-->
[between the differential Galois group (acting on the right) of a differential equation of type \mref{diff_eq2}]

Remark : The derivations $d_x$ cannot in general be expressed by derivations of $\H$. For example, with equation \mref{diff_eq3} <<<mettre diff_eq3 en étiquettede l'équation 40>>>, one has $\delta_{x_0}(\frac{log(z)^{n+1}}{(n+1)!})=\frac{log(z)^{n}}{n!}$ but 
$\delta_{x_0}(\scal{S}{ux_1})=0$. 



===========
Merci Gérard,

Je te mets également des remarque (de forme) de la version 15 :

1) la \mref{LES} à la page 7, après la formule (51) est inesistant.

2) il faute mettre l'environnnement

\begin{enumerate}[i)]
\item ...
\begin{enumerate}

dans les propositions 1 et 2.

3) la \subsection{Through the looking glass: passing from right to left.}
est unique dans ce texte (il n'y a pas d'autre \subsection{}).

Amitiés,
Minh
============
[0] STYLE: (I assume that this is the llncs.cls supplied?)The Author/Institution style may NOT be as required;

also the running head for authors does not work for several  authors.

 

[1] Missing label  \mref{LSE}.  Please supply.

 

[2] After 18, may I replace(coeffcients taken at letters and the empty word)

by (coeffcients of letters and the empty word)

 

[3] In the Conclusions, I believe that some words are needed after

"In this paper we showed that by using fields of germs, some difficult results can be considerably simplified and extended. For instance, polylogarithms were known to be independant over either $\C[z,1/z,1/(1-z)]$ or presumably over "functions which do not bring monodromy"; these two results are now encompassed by Theorem.

It would help, if some of  these results were very briefly indicated.

 

[4] I was not able to find the Condition PI referred to after Eq (28).

 

Please supply your annotations and/or edits s based on this version.

 

Allan


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