\cdot\cdot\cdot > \bR_s$
(as in the chain $C_1$ of the above example) {\em or}
$\bR_1<\cdot\cdot\cdot < \bR_s$ (as in $C_2$).
\erem
%
Finally, we are ready to define {\em complete families of trees
generated by a tree $T$ and a given $T$--admissible function $\a$.}
\dfntwo{Complete families}{dfn:13}
Given $T\in\cT^k$, $\a$ $T$--admissible and $\l<1/2$ we define
\beq
\cF(T)\=\{ T\}
\label{defcF1}
\eeq
if $\cR(T)=\emptyset$ (\ie if there are no $\l$--critical resonances
in $T$), otherwise, letting $\cR\=\cR(T)$, $\cC\=\cC(T)$, we define
\beqa
\cF(T)& \=
\{ T'\in\cT^k: & \ T'= \bT\cup
\bigcup_{\bR\in\bcR} \bR + \nonumber\\
& &
\sum_{C=(\bR_1,...,\bR_s)\in\cC}
u(C)\bu_1+ w(C)\bw_s +
\sum_{i=1}^{s-1} \bw_i\bu_{i+1} \ ,\nonumber\\
& &{\rm where}\ \bu_i,\bw_i\ {\rm vary\ in\ }
\bR_i \}
\label{defcF}
\eeqa
if $s=1$ for some $C$ the sum over $i$ has to be omitted.
$\cF$ {\em is called the complete family associated to
$(T,\a)$}.
\edfn
%
Obviously, the integers $n_i\=\a_{v_i}$,
which together with the ``topological"
structure of the tree $T$ generates $\cF$, may be thought of as fixed
``attributes" of the labels $v_i$ and,
{\em by construction,
$\{\a_{v_i}\}$ is admissible for any tree $T'\in\cF(T)$}.
The following Proposition collects a few elementary properties
of complete families.
\begin{figure}[hbt]
\begin{center}
\begin{picture}(385,340)
\multiput(5,20)(0,80){4}{\begin{picture}(380,80)
\multiput(0,0)(95,0){4}{\begin{picture}(90,80)
\thicklines
\put(20,50){\circle*{3}}
\put(40,40){\line(0,1){20}}
\multiput(40,40)(0,20){2}{\circle*{3}}
\put(60,50){\circle*{3}}
\thinlines
\put(40,45){\oval(26,40)}
\put(40,42.5){\oval(60,60)}
\put(46,18){$R_2$}
\put(65,10){$R_1$}
\put(15,40){$v_1$}
\put(35,30){$v_2$}
\put(42,48){$v_3$}
\put(55,40){$v_4$}
\end{picture}}
\put(0,0){\begin{picture}(380,80)
\thicklines
\put(20,50){\line(2,-1){20}}
\put(40,60){\line(2,-1){20}}
\put(115,50){\line(2,1){20}}
\put(135,60){\line(2,-1){20}}
\put(210,50){\line(2,-1){20}}
\put(230,40){\line(2,1){20}}
\put(305,50){\line(2,1){20}}
\put(325,40){\line(2,1){20}}
\end{picture}}
\end{picture}}
\multiput(5,260)(95,0){4}{\begin{picture}(90,80)
\put(20,50){\circle{6}}
\put(60,50){\line(1,0){20}}
\put(80,50){\circle*{3}}
\put(75,40){$v_5$}
\end{picture}}
\multiput(5,180)(95,0){4}{\begin{picture}(90,80)
\put(20,50){\circle{6}}
\put(0,50){\line(1,0){20}}
\put(0,50){\circle*{3}}
\put(0,40){$v_5$}
\end{picture}}
\multiput(5,100)(95,0){4}{\begin{picture}(90,80)
\put(60,50){\circle{6}}
\put(60,50){\line(1,0){20}}
\put(80,50){\circle*{3}}
\put(75,40){$v_5$}
\end{picture}}
\multiput(5,20)(95,0){4}{\begin{picture}(90,80)
\put(60,50){\circle{6}}
\put(0,50){\line(1,0){20}}
\put(0,50){\circle*{3}}
\put(0,40){$v_5$}
\end{picture}}
\parbox{15cm}{\caption{\label{tab:1} A complete family $\cF$ (Example 1)}}
\end{picture}
\end{center}
\end{figure}
%
\begin{figure}[hbt]
\begin{center}
\begin{picture}(385,340)
\multiput(5,20)(0,80){4}{\begin{picture}(380,80)
\multiput(0,0)(95,0){4}{\begin{picture}(90,80)
\thicklines
\put(20,40){\line(0,1){20}}
\multiput(20,40)(0,20){2}{\circle*{3}}
\put(50,40){\line(0,1){20}}
\multiput(50,40)(0,20){2}{\circle*{3}}
\put(70,50){\circle*{3}}
\thinlines
\put(20,45){\oval(26,40)}
\put(50,45){\oval(26,40)}
\put(10,10){$R_1$}
\put(40,10){$R_2$}
\put(15,30){$v_1$}
\put(8,48){$v_2$}
\put(45,30){$v_3$}
\put(52,49){$v_4$}
\put(65,40){$v_5$}
\end{picture}}
\put(0,0){\begin{picture}(380,80)
\thicklines
\put(20,60){\line(1,0){30}}
\put(115,60){\line(3,-2){30}}
\put(210,40){\line(3,2){30}}
\put(305,40){\line(1,0){30}}
\end{picture}}
\end{picture}}
\multiput(5,260)(95,0){4}{\begin{picture}(90,80)
\put(20,40){\circle{6}}
\put(50,60){\line(2,-1){20}}
\end{picture}}
\multiput(5,180)(95,0){4}{\begin{picture}(90,80)
\put(20,40){\circle{6}}
\put(50,40){\line(2,1){20}}
\end{picture}}
\multiput(5,100)(95,0){4}{\begin{picture}(90,80)
\put(20,60){\circle{6}}
\put(50,60){\line(2,-1){20}}
\end{picture}}
\multiput(5,20)(95,0){4}{\begin{picture}(90,80)
\put(20,60){\circle{6}}
\put(50,40){\line(2,1){20}}
\end{picture}}
\parbox{15cm}{\caption{\label{tab:2} A complete family $\cF$ (Example 2)}}
\end{picture}
\end{center}
\end{figure}
%
\protwo{Properties of $\cF$}{pro:procF}
Let $T\in\cT^k$, $\a$ a $T$--admissible function and $\l<1/2$. Then:
\beqano
& {\rm (i)} & \cF(T)=\{ T\} \sse \ \cR(T)=\emptyset\ ;\\
%
& {\rm (ii)} & \cF(T') = \cF(T)\qquad \forall\ T'\in\cF(T)\ ;\\
%
& {\rm (iiii)} & \cF(T)\cap\cF(T')\neq \emptyset \implies\
T'\in \cF(T)\ ;\\
%
& {\rm (iv)} & \bT,\cR(T),U(T)\quad{\rm is\ a\ complete\
and \ minimal}\\
& &{\rm set\ of\ invariants\ for\ \cF(T)}\ ;\\
%
& {\rm (v)} & |\cF|=\prod_{\bR\in\bcR} |V(\bR)|^2\ ;\\
%
& {\rm (vi)}& \deg_{T'} v\le \deg_{T''} v+2
\qquad \forall\ T',T''\in\cF(T)\\
\eeqano
\epro
The proof of the Proposition is a simple consequence of
the various definitions.
\rem{rem:5.6}
>From (i), (ii), (iii) above it follows immediately that the property
$T'\in\cF(T)$ is an {\em equivalence relation}.
Thus the set of all labeled rooted trees, given
$\{n_i\}\=\{\a_{v_i}\}$, can be {\em partitioned
into disjoint complete families}: here we have preassigned
the function $\a_{v_i}\=n_i$ and we
use the convention that if $\a$ is not admissible for some $T$
we are omitting the (meaningless) contribution coming from that tree.
\erem
%
A final comment: the set of labels $\{v_1,...,v_k\}$ of $\cT^k$
can be thought of as the set of vertices $V$ of $\cF$ and for any
$v\in V$ we set
\beq
\deg_\cF v= \max_{T'\in\cF} \deg_{T'} v
\label{defdegF}
\eeq
and from (vi) of Proposition \ref{pro:procF} it follows immediately that:
\beq
\deg_\cF v - 2 \le \deg_{T'} v\le \deg_\cF v\ ,\qquad \forall\ T'\in\cF
\label{degF}
\eeq
In Figure~\ref{tab:1} and Figure~\ref{tab:2} we show the complete
families associated to order 5 trees with two resonances.
%
\nin
In Figure~\ref{tab:1} we have
$\cR_0=R_1$, $\cR=\{R_1,R_2\}$, $C_1 \equiv \overline{R}_1$, $C_2 \equiv
R_2$, $\overline{T} = \{\eta\} \cup \{v_5\}$, $\overline{R}_1 \equiv \{v_1\}
\cup \{v_4\}$, $u(C_1)=\eta$, $w(C_1)=v_5$, $u(C_2)=v_1$, $w(C_2)=v_4$,
$|\cF|=|V(\bR_1)|^2 |V(R_2)|^2=16$.
\giu
In Figure~\ref{tab:2} we have
$\cR_0=\cR=\{ R_1, R_2\}$, $C_1=(R_1,R_2)$, $u(C_1)=u(R_1)=\eta$,
$w(C_1)=w(R_2)=v_5$,
$|\cF|=|V(R_1)|^2 |V(R_2)|^2=16$.
%********************************************
\newpage
\section{Estimates}
\label{sec:estima}
\setcounter{equation}{0}
%
In this section we formulate the estimates on (sums)
of products of small divisors needed to prove Kolmogorov's
(KAM) theorem following Siegel's strategy. The first two
lemmas are a simple generalization of Siegel's argument
and deal with situations without resonances (or better
with non critical resonances). The first lemma is taken from
\cite{E}. The third and fourth lemmas are the crucial point
of our paper: it is shown that sums over complete families
obey the same bounds that hold for products of small divisors
without resonances (\ie without repetitions). In other words,
the ``divergent terms" whose occurrence has been discussed
in Section~\ref{sec:diverge} compensate (and in fact {\em they
do not cancel exactly}) within complete families.
The four lemmas will easily yield Kolmogorov's theorem for the
case under consideration \equ{he}; see Theorem~\ref{thm:kam}
below. The proofs are presented in the next section.
\giu
Recall \equ{dioph} and the definition of admissible functions
\equ{adm}.
%
\lemtwo{Siegel, Eliasson}{lem:se}
Let $T\in\cT^k$ be a rooted tree of order $k$ (\ie
$k$ $\=$ cardinality of $V(T)$), $\a$ a $T$--admissible
function and $0<\l<1$. Assume that
\beq
{\rm if}\ \d_u=\d_w\ {\rm for\ some\ }u>w \ \ {\rm then\ }
\exists\ v\ {\rm s.t.} \ u>v>w\ {\rm and}\ |\d_v|>\l |\d_u|
%
\label{hy1}
\eeq
Then there exist constants $c_1>\g$, $\b_1>\t$ such that
\beq
\prod_{v\in T} |\d_v|^{-1}\le c_1^k \ \prod_{v\in T} |\a_v|^{\b_1}
\label{e.1}
\eeq
The constant $c_1,\b_1$ can be taken to be
\beq
c_1=\g\ 2^{6\t}\ \frac{1+\l}{\l}\ ,\qquad
\b_1=3\t
\label{const1}
\eeq
\elem
%
Next Lemma extend the above type of estimate to
products of small divisors arising in trees with (possible)
resonant subtrees which are non $\l$--resonant (for some
prefixed $\l<1/2$); recall that such trees may contain couples
of points which violate
\equ{hy1} (see Remark~\ref{rem:5.1})
so that the result described in the next Lemma
is a genuine generalization of the estimates described in
Lemma~\ref{lem:se}.
%
\lem{lem:2}
Let $T$ and $\a$ be as in Lemma~\ref{lem:2} and let
$0<\l<1/2$.
Assume that there are no $\l$--resonance (Definition~{\rm\ref{dfn:4}}).
Then there exist constants $c_2>\g$ and $\b_2>\t$ such that
\beq
\prod_{v\in T} |\d_v|^{-1}\le c_2^k \ \prod_{v\in T} |\a_v|^{\b_2}
\label{e.2}
\eeq
The constant $c_2,\b_2$ can be taken to be
\beq
c_2=c_1\ 2^{\t}\ \frac{1-\l}{1-2\l}\ ,\qquad
\b_2=\b_1+\t
\label{const2}
\eeq
\elem
%
\lemtwo{Small divisor compensations}{lem:3}
\HB\nin
Let $T$, $\a$ and $\l$ be as in Lemma~{\rm\ref{lem:2}} and
let $0<\bl<1$.
Let $R$ be a minimal $\l$--(sub)resonance of $T$
(\ie $R$ does not contain any $\l$--subresonance,
see Definition~{\rm\ref{dfn:7}}) and
let $n\in\zNn$ be such that $\agl{n}\le \bl \D(R)$.
Let $z$ be an extra vertex (not in $T$) and
set $\a_z=n$. Finally, for any $u,w\in R$,
denote by $R_u^w$ the tree\footnote{\rm If $R$ is rooted,
$R=R_r$, first remove the edge $\eta r$ so as to obtain
an unrooted subtree.}
$R\cup \{z\}+wz$ rooted at $u$.
Then there exist constants $c_3>\g^2$, $\b_3>2\t$ such that
for all $1\le i,j\le N$
\beq
|\sum_{u,w\in R} \a_{ui} \a_{wj}
\prod_{v\in R} \d_v(R_u^w)^{-2}|
\le c_3^{|V(R)|} \ \prod_{v\in R} |\a_v|^{\b_3}
\label{e.3}
\eeq
where $\a_{ui}$ (respectively $\a_{uj}$) denote
the $i^{\rm th}$ (respectively the $j^{\rm th}$)
component of the integer vector $\a_u$.
The constant $c_3,\b_3$ can be taken to be
\beq
c_3=\big( c_2\ 2^{\t+1} (1-\bl)^{-1} \big)^2\ ,\qquad
\b_3=2(\b_2+\t+1)
\label{const3}
\eeq
\elem
%
The estimates \equ{e.3} lead easily to control the
contribution to \equ{exptree} coming from
complete families.
%
\lem{lem:4}
Let $T$, $\a$ and $\l<1/2$ be as above. Let $\cF\=\cF(T)$ be
the complete family defined in Definition~{\rm\ref{dfn:13}},
$V\=V(T)=V(\cF)$ and recall the definition of $\deg_\cF$
given in \equ{defdegF}. Then there exist constants
$c_4>\g^2$, $\b_4>2\t$ such that
\beq
\big|
\sum_{T'\in\cF} \prod_{vv'\in E(T')} \a_v\cdot\a_{v'}
\prod_V \d_v^{-2}\big| \le
\prod_{v\in V} |\a_v|^{\deg_\cF v} \ c_4^{|V|}\ \prod_{v\in V}
|\a_v|^{\b_4}
\label{e.4}
\eeq
The constants $c_4,\b_4$ can be taken to be
\beq
c_4\=(c_2\ 2^{\t+1}\ \frac{1-\l}{1-2\l} )^2\ ,
\qquad \b_4= \b_3
\label{const4}
\eeq
\elem
%
Kolmogorov's theorem follows now easily. We introduce the following
norms on analytic functions $g:\tN\to\complex$:
\beq
\| g\|_\s \= \sum_{n\in\zN} |g_n| e^{|n|\s}
\label{norm}
\eeq
Clearly, for any analytic function on $\tN$ there exist a $\s>0$ such
that the above norm is finite; viceversa if $g$ is such that
$\|g\|_\s<\infty$ for some $\s>0$ then $|g_n|\le \|g\|_\s \exp(-|n|\s)$
so that $g$ is analytic. If $g=(g_1,...,g_M):\tN\to\complex^M$
we set
\beq
\|g\|_\s\=\sup_{1\le j\le M} \|g_j\|_\s
\label{norm2}
\eeq
%
\thm{thm:kam}
Let $f$ in \equ{he} be real--analytic with $\| f\|_\bs<\infty$
for some $\bs>0$. Then the formal solution $X(\th,\e)$
described in Proposition~{\rm\ref{pro:ef}} is real--analytic.
Furthermore, fix $0<\s<\bs$, $0<\l<1/2$ and set
\beq
\e_0\=\frac{(\bs-\s)^{\b_5}}{c_5 \|f\|_\bs}\ \quad
{\rm with}\quad\cases{ \b_5\=\b_4+4\cr
c_5\= c_4 2^{\b_5+1}\ \b_5!\cr}
\label{const5}
\eeq
where the constants $\b_4,c_4$ are defined above.
Then $X(\th,\e)$ is jointly analytic in the domain
$\{|\Im \th_i|\le \s\}$ $\times$ $\{|\e|\le \e_0\}$ and for
any (complex) $\e$ with $|\e|<\e_0$ the following bound holds
\beq
\|X(\cdot,\e)\|_\s \le \frac{\bs-\s}{2}\
\frac{|\e|}{\e_0-|\e|}
\label{est}
\eeq
\ethm
%
\rem{rem:constants} Choosing $\l=1/5$ (so as to minimize the constant
$c_5$) one obtains
\beq
\b_5= 10\t+12\ ,\qquad c_5< \g^2 \ 2^{26 \t+23}\ (10\t+12)!
\eeq
Such constants are certainly not optimal.
\erem
%
%********************************************
\section{Proofs}
\label{sec:proofs}
\setcounter{equation}{0}
%
In the following five Subsections we give
the proofs of the four Lemmas
and of the Theorem of Section~\ref{sec:estima}.
In the first Subsection we show how Kolmogorov's Theorem
can be easily derived from Lemma~\ref{lem:4} and in the remaining
Subsections we give the proofs of the four Lemmas.
%
\subsection{Proof of Theorem {\bf 6.1}} %\ref{thm:kam}
\label{subsec:1}
%
Recall the form of the coefficients $\Xk_{jn}$ given
in \equ{exptr1} $\div$ \equ{exptr3}.
By Remark~\ref{rem:5.6}
we know that, for a given choice of $\{n_1,...,n_k\}$
with $n_i\in\zNn$,the labeled rooted trees in $\cT^k$ can be
partitioned into complete families having as common
admissible $\a$ function $\a_{v_i}=n_i$. Denote by
$\calN\=\calN(n_1,...,n_k)$ the set of all such complete
families. Then, by \equ{exptr1} $\div$ \equ{exptr3},
by Lemma~\ref{lem:4} and by \equ{defdegF}, \equ{degF}
we see that, for any $0<\s<\bs$,
\beqa
&&\sum_{n\in\zNn} e^{\s|n|} |\Xk_{jn}| \nonumber\\
&&= \sum_n e^{\s|n|} \frac{1}{k!}\ \Big|
\sum_{ {n_1,...,n_k}\atop{n_i\neq 0,\sum n_i=n}}
\sum_{\cF\in\calN} \prod_{i=1}^k f_{n_i}
\sum_{T\in \cF} \prod_{vv'\in E(T)} \a_v\cdot\a_{v'}
\prod_{V} \d_v^{-2}\Big| \nonumber\\
&& \le \sum_{n} \frac{1}{k!}
\sum_{ {n_1,...,n_k}\atop{n_i\neq 0,\sum n_i=n}}
\sum_{\cF\in\calN} \prod_{i=1}^k \big(f_{n_i}e^{\s|n_i|}\big)
\prod_{v\in V} |\a_v|^{\deg_\cF v} c_4^k
\prod_{V} |\a_v^{\b_4}|\nonumber\\
&&\le
\frac{1}{k!} \sum_{T\in\cT^k} \sum_{\a:T\to\zN}
\prod_{V} \big( c_4 |f_{\a_v} |\a_v|^{\b_4+2} e^{\s |\a_v|}\big)
\prod_V |\a|^{\deg v}
\label{6.1}
\eeqa
In view of Proposition~\ref{pro:exptree2} we see that, if we define
\beq
\phi_n\=c_4|f_n| e^{\s|n|} |n|^{\b_4+2}
\label{6.2}
\eeq
then, last line of \equ{6.1} is exactly the formal solution
$g$ of the equation \equ{gphi} with coefficient $g_k$
given in \equ{exptree2}.
But since (by the analyticity of $f$) the nonnegative numbers
$\phi_n$ decay exponentially fast as $|n|\to\infty$,
the function of two complex variables
\beq
G(x,\e)\= x-\e\sum_{n\in\zN} \phi_n |n| e^{|n|x}
\label{6.3}
\eeq
is holomorphic and bounded in the complex $2$--disk
\beq
D\=\{ x\in\complex:\ |x|\le s\} \times
\{\e\in\complex: \ |\e|\le\e_0\}
\label{6.4}
\eeq
for any $\e_0>0$ and any $0 From now on we assume that $I_1\neq \emptyset$ \ie $m\ge p_1>0$.
We shall adopt the convention that {\em sums or products
over empty set of indices have to be disregarded}
(\ie replaced, respectively, by $0$ or $1$).
Next, let
\beq
\bar \a_{u_i}\=\a_{u_i}+\a(S_{ij})\ \qquad \implies \quad
|\bar \a_{u_i}|\le x_i+y_i
\label{6.33}
\eeq
then $\d_{u_i}\=\d_{u_i}(T;\a)=\d_{u_i}(P;\bar \a)$, whence
\beq
\prod_T |\d_v|^{-1}= \prod_P |\d_{u_i}(P;\bar \a)|^{-1}
\prod_{i\in I_1} \prod_{j=1}^{m_i} \prod_{v\in S_{ij}}
|\d_v|^{-1}
\label{6.34}
\eeq
and we can use Sublemma~\ref{sublem:1} to bound the product
over $P$ (as $(P,\bar \a)$ obviously satisfy \equ{6.8})
and the inductive hypotheses to bound the product over $S_{ij}$
(note that $\d_v(T)=\d_v(S_{ij}$)).
Recalling the definitions in \equ{defvarie}, we obtain from
\equ{6.34} and \equ{6.33}
\beq
\prod_{v\in T} |\d_v|^{-1} \le
M B D^{k-1} \big(\prod_T |\a_v|^{3\t}\big) |\a|(T)^{-2\t}
\label{6.35}
\eeq
with
\beq
M\= \frac{B_0}{B} \big(\frac{B}{D}\big)^m
\big(\frac{D_0}{D}\big)^p \big[ \prod_{I_0}
x_i^{-2}\cdot\prod_{I_1}\big(\frac{x_i+y_i}{x_i^3}
\prod_jy_{ij}^{-1}\big)\cdot|\a|(T)^2\big]^\t
\label{6.34bis}
\eeq
and we have to show that $M\le 1$ if we choose suitably
$D,B$.
Suppose first that $p\in I_0$. Then by
the first inequality in \equ{6.31} with $i=p$ we have
\beq
x_p>\frac{1}{2} |\a|(T)
\label{6.35bis}
\eeq
Then using repeatedly the bound $\sum_{i=1}^m b_i\le m\prod_{i=1}^m b_i$
$\le$ $2^{m-1} \prod_{i=1}^m b_i$ (valid for real numbers $b_i\ge 1$)
and estimating $|\a|(T)^2$ in \equ{6.34bis} by $4x_p^2$ we get
\beq
M\le \frac{B_0}{B} \big(\frac{D_0}{D}\big)^p
\big[ \prod_{{i\in I}\atop{i\neq p}} x_i^{-2}
2^{p_1+2}\ 2^{m-p_1}\ \prod_{i\in I_1,j} y_{ij}^{-1}
\big]^\t\le 1
\label{6.36}
\eeq
provided
\beq
D\ge 2^\t \max\{B,2^\t D_0\}\ ,\qquad B\ge B_0 2^{2\t}
\label{6.37}
\eeq
(which is stronger than \equ{6.32}).
Suppose now that $p\in I_1$. We can assume that $x_p\le
|\a|(T)/2$ (otherwise \equ{6.35bis} holds also in this
case and we can proceed as above). Then
\beq
M\le \frac{B_0}{B} \big(\frac{B}{D}\big)^m
\big(\frac{D_0}{D}\big)^p \big[ 2^{2m}\
\prod_{{i\in I}\atop{i\neq p}}x_i^{-1}
\cdot\prod_{{i\in I_1}\atop{i=neq p}}
y_i^{-1} \cdot x_p^{-2}
\prod_{j=1}^{m_p} y_{pj}^{-2}\cdot y_p
|\a|(T)^2\big]^\t
\label{6.38}
\eeq
and we see that we are in position to apply Sublemma~\ref{sublem:2}
(with $r=p+p_1-1$, $s=m_p+1$, $z=|\a|(T)=\sum_I x_i+\sum_{I_1} y_i$)
obtaining
\beq
M\le \frac{B_0}{B} \big(\frac{B}{D}\big)^p
\big(\frac{D_0}{D}\big)^p \big[ 2^{2m}\ 2^{p+p_1-1}\
2^{2(m_p+1)}\ \frac{y_p}{|\a|(T)}\big]^\t\le 1
\label{6.39}
\eeq
provided we take
\beq
B\= B_0 \ 4^\t\ ,\qquad
D\= 4^\t \max\{ D_0,B_0 2^{4\t} \}
\label{6.40}
\eeq
(which satisfy also \equ{6.37}). Thus, by \equ{6.10},
we can take
\beq
\b_1\=3\t\ ,\qquad c_1=\g\ 2^{6\t}\ \frac{1+\l}{\l}\
\label{6.41}
\eeq
This concludes the proof of Lemma~\ref{lem:se}. \qed
%
\subsection{Proof of Lemma {\bf 6.2} } %\ref{lem:2}
\label{subsec:3}
%
We prove the Lemma by induction on $k\=|V|$. For $k=1$,
\equ{e.2} follows from \equ{dioph} (as $c_2>\g$ and $\b_2>\t$).
Let $k\ge 2$, assume that the Lemma holds for all
trees with up to $k-1$ vertices and let $T\in\cT^k$
have no $\l$--resonances.
Recall that even though $T$ does not contain $\l$--resonances,
it may contain either $\l$--resonant couples of points
(compare Remark~\ref{5.1}), or {\em adjacent points} $u,w$
such that $\d_u=\d_w$ (recall that the main hypothesis
of Lemma~\ref{lem:se}, \equ{hy1}, excludes the presence
of adjacent points of constancy for $\d$).
For any couple $(u,w)$ of $\l$--resonant points let
$R(u,w)$ be the associated resonant subtree (see
\equ{5.3}).
Let $R$ be a minimal resonant tree associated to a couple
of $\l$--resonant points (minimal meaning that $R$
does not contain $\l$--resonant points); if there are
no $\l$--resonant points we set $R=\emptyset$.
There are two cases:
\nin
(i) either $R=\emptyset$ or $R$ contains two adjacent points
$u>w$ such that $\d_u=\d_w$, or
\nin
(ii) $R\neq \emptyset$ and $R$ does not contain any couple
of adjacent points on which $\d$ is constant.
\nin
Consider case (i): if there are no couples of adjacent points
$u,w$ such that $\d_u=\d_v$, then Lemma~\ref{lem:2} follows from
Lemma~\ref{lem:se}, otherwise let $S=S(v_1,v_2)$ be a minimal
resonant subtree such that $v_1$ and $v_2$ are adjacent and
of constancy for $\d$ (minimal means that $S$ does not contain
any couple of adjacent points on which $\d$ is constant).
Since $\a_v\neq 0$ for all vertices $v$, $S$ contains at least two
points and by hypothesis $S$ is not $\l$--resonant. Thus observing
that
\beq
\prod_T |\d_v|^{-1} = \prod_{T/S} |\d_v|^{-1} \prod_S |\d_v|^{-1}
\= \prod_{T/S} |\d_v|^{-1} \prod_S |\d_v(S_{v_1})|^{-1}
\label{6.42}
\eeq
we see that we can apply Lemma~\ref{lem:se} to the tree $S_{v_1}$
while to the tree $T/S$ we can use the inductive
%FOOTNOTE
hypothesis\footnote{Note that contracting resonant trees associated to
adjacent points where $\d$ is constant cannot introduce new resonances,
thus all resonances in $T/S$ are not $\l$--resonant.}
%
so that \equ{e.2} holds in case (i) provided
\beq
c_2\ge c_1\ ,\qquad \b_2\ge \b_1
\label{6.43}
\eeq
%
Consider case (ii).
It is easy to see that the tree $T/R$ does
not contain any $\l$--resonance and since
\beq
\prod_T |\d_v|^{-1} = \prod_{T/R} |\d_v|^{-1} \prod_R |\d_v|^{-1}
\label{6.44}
\eeq
we see that we can estimate, by the inductive hypothesis,
the product over $T/R$. To bound the product over $R=R(u,w)$,
let $n\=\sum_{v'\le w} \a_{v'}$. Then, since $(u,w)$ form
a couple of $\l$--resonant points, for any $u~~w\ \ {\rm then}\ \exists
\ v \ {\rm between} \ u\ {\rm and}\ w \ {\rm s.t.}\
|\d_w|>\l|\d_v|
\label{6.8}
\eeq
Then there exist positive constants $D_0\ge B_0\ge \g$ such that
\beq
\prod_P |\d_v|^{-1}\le B_0 D_0^{k-1} \prod_V |\a_v|^\t
\label{6.9}
\eeq
The constants $B_0,D_0$ can be taken to be
\beq
B_0=\g\ ,\qquad D_0=\g\ 2^\t \ \frac{1+\l}{\l}
\label{6.10}
\eeq
\esublem
%
\PROOF By induction on $k$. If $k=1$, \equ{6.9}
follows at once from the Diophantine inequality \equ{dioph}.
Let $k\ge 2$ and assume the statement true for all rooted paths
with up to $k-1$ vertices satisfying \equ{6.8}. Let $(P,\a)$ as
above. Define $s$ to be the greatest integer between $1$ and $k$
such that
\beq
|\d_{v_s}|\ge \max_{1\le i\le k} |\d_{v_i}|
\label{6.11}
\eeq
There are four cases:
\beqa
& {\rm (i)} \ s=1 \ , &{\rm(ii)}\
\a_{v_s}+\a_{v_{s-1}}\neq 0\ ,
\label{6.12} \\
&{\rm (iii)} \ \a_{v_s}+\a_{v_{s-1}}= 0\ , s+1~~