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\begin{document}
\section{Amalgamation of valuations}
\begin{Theorem}\label{Theoremg1}
Suppose that $P_i\in K[X,Y]$ are defined for $i\in \NN$ by $P_0=X$ $P_1=Y$,
$$
P_{i+1}= \lambda_i P_i^{m_i}-\lambda_{i,0}M_i-\sum_{\ell_i'}\lambda_{i,\ell_i'}M_{i,\ell_i'}
$$
for $i\ge 1$, where $\ell_i'=(\ell_{i,0}',\ldots,\ell_{i,i}')$,
$M_i=\prod_{i'=0}^{i-1} P_{i'}^{\ell_{i,i'}}$, 
$M_{i,\ell_i'}=\prod_{i'=0}^{i} P_{i'}^{\ell_{i,i'}'}$
with $0\le \ell_{i,i'},\ell_{i,i'}<m_{i'}$ for $1\le i'\le i$, $\ell_{i,0},\ell_{i,0}'\ge 0$
arbitrary, and $\lambda_i, \lambda_{i,0},\lambda_{\ell_i'}\in K$ with $\lambda_i, \lambda_{i,0}\ne 0$.

Define $\gamma_0=1$ and by induction on $i$, $\gamma_i=\frac{1}{m_i}\sum_{i'=0}^{i-1}\ell_{i,i'}\gamma_{i'}$.
Let $\Gamma_i =< \{{\gamma_{i'}}\}_{0\le i'\le i}>$ be the group generated by
$\gamma_{i'}$ for $0\le i'\le i$. Suppose that
\begin {enumerate}
 \item $\sum_{i'=0}^{i-1}\ell_{i,i'}\gamma_{i'}$ has
 order $m_i$ in $\frac{\Gamma_{i-1}}{m_i\Gamma_{i-1}}$ for all $i\ge 1$,
\item $\gamma_{i+1}>m_i\gamma_i$ for all $i\ge 1$, and
\item $\sum_{i'=0}^{i}\ell_{i,i'}'\gamma_{i'}>m_i\gamma_i$ for all $\ell_i'$.
\end {enumerate}

Then
\begin{enumerate}
 \item  Suppose that $f\in K[X,Y]$ and $\ell$ is such that $\mbox{deg}_y f<\prod_{j=1}^
{l-1}m_i$. Then $f$ has a unique finite expansion
\begin{equation}\label{g1}
f= \sum_{\alpha}a_{\alpha}\prod_{i=0}^{\ell}P_i^{\alpha_i}
\end{equation}
with all $\alpha=(\alpha_0,\ldots,\alpha_{\ell})\in \NN^{\ell+1}$, $0\le \alpha_i<m_i$ for $1\le i$ and
$a_{\alpha}\in K$,
\item
In the expansion $(\ref{g1})$, The numbers
$\sum_{i=0}^{\ell}\alpha_i\gamma_i$ are distinct for distinct $\alpha$,
\item The rule
$$
\nu_1(f)=\mbox{min}_{\alpha} \sum_{i=0}^{\ell}\alpha_i\gamma_i
$$
where the minimum is computed for the expansion (\ref{g1}), determines a nondiscrete
rational rank 1 valuation $\nu_1$ of $K(X,Y)$, dominating the local ring $K[X,Y]_{(X,Y)}$.
\end{enumerate}


\end{Theorem}

\begin{Theorem}\label{Theoremg2} Suppose that $R$ is a local  domain with quotient field $K$ and  $\overline \nu$ is a valuation of $K$ which dominates $R$, with value group $\overline\Gamma$. Suppose that $\{P_i\}$ is a sequence of polynomials in $K(X,Y)$ satisfying the assumptions of Theorem \ref{Theoremg1}. Define by induction on $i$,
$\tau_0=0$, and 
$$
\tau_i=\frac{1}{m_i}\sum_{i'=0}^{i-1}\ell_{i,i'}\tau_{i'}+\overline\nu(\lambda_{i,0})
-\overline\nu(\lambda_i)
\in \overline\Gamma_{\QQ}=\overline\Gamma\otimes_{\bf \ZZ}{\bf \QQ}.
$$
For $f\in K[X,Y]$, define from the expansion (\ref{g1}),
$$
\nu(f) =\mbox{min}_{\alpha}\{ \sum_{i=0}^{\ell}\alpha_i(\gamma_i,\tau_i)+(0,\mbox{ord}_z(a(z))\}\in (\QQ\times \overline\Gamma_{\QQ})_{\mbox{lex}}.
$$
Then $\nu$ defines a rank 2 valuation on $K(X,Y)$ which is the composition of $\nu_1$ and $\overline \nu$. Further, $\nu$ dominates the local ring $A=R[X,Y]_{m_RR[X,Y]+(X,Y)}$,
and the center of $\nu_1$ on $A$ is the prime ideal $(X,Y)$.


\end{Theorem}


\begin{Theorem}\label{Theoremg3} Suppose that $R$ is a local  domain with quotient field $K$ and  $\overline \nu$ is a valuation of $K$ which dominates $R$, with value group $\overline\Gamma$. Suppose that $\{P_i\}$ is a sequence of polynomials in $K(X,Y)$ satisfying the assumptions of Theorem \ref{Theoremg1} and
$\{Q_i\}$ is a sequence of polynomials in $K(U,V)$ satisfying the assumptions of Theorem \ref{Theoremg1}, where 
$$
P_{i+1}= \lambda_i P_i^{m_i}-\lambda_{i,0}M_i-\sum_{\ell_i'}\lambda_{i,\ell_i'}M_{i,\ell_i'}
$$
and
$$
Q_{i+1}= \mu_i Q_i^{n_i}-\mu_{i,0}N_i-\sum_{\ell_i'}\mu_{i,\ell_i'}N_{i,\ell_i'}
$$

Let $\nu_1$ be the valuation on $K(X,Y)$ defined by the $\{P_i\}$, with  $\nu_1(P_i)=\gamma_i$, and let 
$\nu_2$ be the valuation on $K(U,V)$ defined by the $\{Q_i\}$, with  $\nu_2(Q_i)=\delta_i$.



Define by induction on $i$,
$\tau_0=0$, and 
$$
\tau_i=\frac{1}{m_i}\sum_{i'=0}^{i-1}\ell_{i,i'}\tau_{i'}+\overline\nu(\lambda_{i,0})
-\overline\nu(\lambda_i)
\in \overline\Gamma_{\QQ}=\overline\Gamma\otimes_{\bf \ZZ}{\bf \QQ},
$$
$\zeta_0=0$, and 
$$
\zeta_i=\frac{1}{n_i}\sum_{i'=0}^{i-1}\ell_{i,i'}\zeta_{i'}+\overline\nu(\mu_{i,0})
-\overline\nu(\mu_i)
\in \overline\Gamma_{\QQ}=\overline\Gamma\otimes_{\bf \ZZ}{\bf \QQ}.
$$



Then
\begin{enumerate}
 \item Suppose that $f\in K[X,Y,U,V]$, with $\mbox{deg}_Yf<\prod_{j=1}^{\ell-1}m_i$
and $\mbox{deg}_Vf<\prod_{j=1}^{m-1}n_i$. 
Then $f$ has a unique finite expansion
\begin{equation}\label{g2}
f= \sum_{\alpha,\beta}a_{\alpha,\beta}\prod_{i=0}^{\ell}P_i^{\alpha_i}
\prod_{i=0}^{m}Q_i^{\beta_i}
\end{equation}
with all $\alpha=(\alpha_0,\ldots,\alpha_{\ell})\in \NN^{\ell+1}$, $0\le \alpha_i<m_i$ for $1\le i$,  all $\beta=(\beta_0,\ldots,\beta_m)\in \NN^{m+1}$, $0\le \beta_i<n_i$ for $1\le i$, and
$a_{\alpha,\beta}\in K$.
\item Suppose that $G$ is a totally ordered abelian group, and $h_1:\QQ\rightarrow G$ and $h_2:\QQ\rightarrow G$ are order preserving embeddings.
For $f\in K[X,Y,U,V]$, define from the expansion (\ref{g2}),
$$
\nu(f) =\mbox{min}_{\alpha,\beta}\{ \sum_{i=0}^{\ell}\alpha_i(h_1(\gamma_i),\tau_i)
+\sum_{i=0}^{m}\beta_i(h_2(\delta_i),\zeta_i)
+(0,\mbox{ord}_z(a(z))\}
\in (G\times \overline\Gamma_{\QQ})_{\mbox{lex}}.
$$
Then $\nu$ defines a  valuation on $K(X,Y)$ which is  composite with $\overline \nu$.
$\nu$ has rank 2 or 3, depending on if the  rank
  of $h_1(\QQ)+h_2(\QQ)$ is 1 or 2. Further, $\nu$ dominates the local ring
 $A=R[X,Y,U,V]_{m_RR[X,Y,U,V]+(X,Y,U,V)}$.


\end{enumerate}




\end{Theorem}



\section{Wild behavior of the tilde function}

\begin{Theorem}\label{Theoremg4}
 Given any decreasing function $f:\NN\rightarrow \ZZ$, there exists a valuation $\nu$
with value group $(\frac{1}{2^\infty}\ZZ\times \ZZ)_{\mbox{lex}}$ which dominates a local ring
$R$ of dimension 3, such that for any valuation $v$ equivalent to $\nu$, with value group $(\frac{1}{2^\infty}\ZZ\times \ZZ)_{\mbox{lex}}$, we have that for all sufficiently large $n\in\NN$, there exists $\lambda\in \frac{1}{2^\infty}\ZZ\cap [0,n]$ such that
 $\pi_2(\tilde\lambda)<f(n)$.
\end{Theorem}

\begin{Theorem}\label{Theoremg5}
 Given any increasing function $g:\NN\rightarrow \ZZ$, there exists a valuation $\omega$
with value group $(\frac{1}{2^\infty}\ZZ\times \ZZ)_{\mbox{lex}}$ which dominates a local ring
$R$ of dimension 3, such that for any valuation $w$ equivalent to $\omega$, with value group $(\frac{1}{2^\infty}\ZZ\times \ZZ)_{\mbox{lex}}$, we have that for all sufficiently large $n\in\NN$, there exists $\lambda\in \frac{1}{2^\infty}\ZZ\cap [0,n]$ such that
 $\pi_2(\tilde\lambda)>g(n)$.
\end{Theorem}

\begin{Theorem}\label{Theoremg6}
 Suppose that $f:\NN\rightarrow \ZZ$ is a decreasing function and 
 $g:\NN\rightarrow \ZZ$ is an increasing function. Then there exists a valuation $\mu$
with value group $((\frac{1}{2^\infty}\ZZ+\frac{1}{2^\infty}\ZZ\sqrt{2})\times \ZZ)_{\mbox{lex}}$ which dominates a local ring
$A$ of dimension 5, such that for any valuation $m$ equivalent to $\mu$, with value group $((\frac{1}{2^\infty}\ZZ+\frac{1}{2^\infty}\ZZ\sqrt{2})\times \ZZ)_{\mbox{lex}}$, we have that for all sufficiently large $n\in\NN$, there exists $\lambda_1\in (\frac{1}{2^\infty}\ZZ+\frac{1}{2^\infty}\ZZ\sqrt{2})\cap [0,n]$ such that
 $\pi_2(\tilde\lambda_1)>f(n)$, and
there exists $\lambda_2\in (\frac{1}{2^\infty}\ZZ+\frac{1}{2^\infty}\ZZ\sqrt{2})\cap [0,n]$ such that
 $\pi_2(\tilde\lambda_2)<g(n)$.
\end{Theorem}



\end{document}



Let $\nu_1$ and $\nu_2$ be valuations on the noetherian local ring $(R,m)$, with centers $p_2\subset_{\ne} p_1$ and such that $\nu_1$ is composed with $\nu_2$ and their ranks differ by one. Let $\lambda\colon \Gamma_1\to \Gamma_2$ be the corresponding map of value groups.  Set $t_1=\nu_1 (p_1), t_2=\nu_2(p_2)$.\par
Given $\phi_2\in \Gamma_2$, denote by $\tilde\phi_2\in \Gamma_1$ the minimum of $\{\nu_1 (f); f\in R,\ \nu_2(f)=\phi_2\}$. Note that $\lambda(\tilde \phi_2)=\phi_2$ and that $t_1\in \hbox{\rm Ker}\lambda$. Since the ranks of the valuations differ by one, the ordered group $\hbox{\rm Ker}\lambda$ is of rank one, and we identify it with a subgroup of the ordered group $\R$.\par
We remark that for $\phi_1\in [\tilde \phi_2,\tilde\phi_2+y_1t_1]$ we have the inclusions\par\noindent
$$p_1^{y_1}\P_{\phi_2}\subset \P_{\phi_1}\subset \P_{\phi_2},$$ and since the valuation $\overline\nu_1$ which is the image of $\nu_1$ in $R/p_2$ is of rank one, the number of elements in the interval $ [\tilde \phi_2,\tilde\phi_2+y_1t_1]$ is finite.\par
The elements of the group $\Gamma_i$ will be denoted by $\phi_i$ or $\gamma_i$, and it is understood that $\P_{\phi_i}=\{x\in R; \nu_i(x)\geq \phi_i\}$.
\begin{lemma}\label{ineq} For any additive function $A$ on $R$-modules we have 
$$\sum_{\phi_1\in[\tilde \phi_2,\tilde\phi_2+y_1t_1]}A(\P_{\phi_1}/\P_{\phi_1}^+)\leq A(M_{\phi_2}/p_1^{y_1}M_{\phi_2}),\leqno(1)$$ where $M_{\phi_2}= \P_{\phi_2}/ \P^+_{\phi_2},$ which is a finitely generated torsion free $R/p_2$-module.\end{lemma}
Indeed, by additivity the left hand side is equal to $A(\frac{M_{\phi_2}}{\P_{\phi_2}/\P_{\tilde \phi_2+ty_1}})$ which is a quotient of $M_{\phi_2}/p_1^{y_1}M_{\phi_2}$.
\qed

In the special case where $\nu_1$ is a valuation of rank one and $\nu_2$ is the trivial valuation, we have $\Gamma_2=0$, $\tilde \phi_2=0\in \Gamma_1$ and the inequality reduces to:
$$\sum_{\phi\in[0,y_1t_1]}A(\P_{\phi}/\P_{\phi}^+)\leq A(R/p_1^{y_1}R),\leqno(2)$$

\par\noindent
We can take as additive function $A$ on $R$-modules of dimension $\leq\hbox{\rm dim}R/p_1$ the multiplicity in dimension $\hbox{\rm dim}R/p_1$ with respect to the maximal ideal.  The inequality $(1)$ then becomes 

$$\sum_{\phi_1\in[\tilde \phi_2,\tilde\phi_2+y_1t_1]}e_m(\P_{\phi_1}/\P_{\phi_1}^+)\leq e_m(M_{\phi_2}/p_1^{y_1}M_{\phi_2}),\leqno(3)$$
It is possible to evaluate the right hand side thanks to the:
\begin{proposition}\label{additivity} {\rm (See ([B], \S7, No.1, Prop 3)}Given a finitely generated $R$-module $M$ of dimension $d\geq 0$, denote by $B$ the set of minimal prime ideals of $\hbox{\rm Supp}(M)$ such that $\hbox{\rm dim}(R/p)=d$. Let $q$ be an ideal of $R$ contained in its radical and such that $M/qM$ has finite length. Then we have 
$$e_q(M)=\sum_{p\in B}\hbox{\rm length}  (M_p)e_q(R/p).$$
\end{proposition}\qed\par\noindent
We shall also use the following
\begin{corollary}\label{semic} Let $(R,m)$ be a noetherian local ring and let $M$ be a finitely generated module whose annihilator is a prime ideal $p$. Let $q$ be a prime ideal of $R$ containing $p$. Then we have the inequality
$$e_{qR_q}(M_q)\leq e_m(M).$$ 
\end{corollary}\par\noindent
By Proposition \ref{additivity}, we have the following two equalities:
$$e_{qR_q}(M_q)=\hbox{\rm length}_{(R_q)_{pR_q}}((M_q)_{pR_q})e_{qR_q}(R_q/pR_q)\ \hbox{\rm and}\ e_m(M)=\hbox{\rm length}_{R_p}(M_p)e_m(R/p).$$
Now it suffices to use the fact that $(R_q)_{pR_q}=R_p$, $(R/p)_{q/p}=R_q/pR_q$ and $(M_q)_{pR_q}=M_p$, and the semicontinuity of mutiplicity (see [L]) which tells us that $e_{qR_q}(R_q/pR_q)\leq e_m(R/p).$ \qed\par\medskip 
 If we apply Proposition \ref{additivity} to $M_{\phi_2}/p_1^{y_1}M_{\phi_2}$ we get:\par\noindent
$$e_m(M_{\phi_2}/p_1^{y_1}M_{\phi_2})= \hbox{\rm length}_{R_{p_1}}\bigl((M_{\phi_2})_{p_1}/m^{y_1}_1(M_{\phi_2})_{p_1}\bigr)e_m(R/p_1),$$
where $m_1=p_1R_{p_1}$ and the support of the $R$-module $M_{\phi_2}$ is the ideal $p_2$.\par\noindent
By the main result on Hilbert-Samuel polynomials, in view of the fact that the support of $M_{\phi_2})_{p_1}$ is the ideal $p_2R_{p_1}$, we have an asymptotic estimate:
$$\hbox{\rm length}_{R_{p_1}}\bigl((M_{\phi_2})_{p_1}/m^{y_1}_1(M_{\phi_2})_{p_1}\bigr)\asymp \frac{e_{m_1}\bigl((M_{\phi_2})_{p_1})\bigr)}{(\hbox{\rm dim}(R_{p_1}/p_2R_{p_1})!}y_1^{\hbox{\rm dim}(R_{p_1}/p_2R_{p_1})}$$
%--------------------------------------------------------------
Let $$(0)=p_{n+1}\subset p_n\subset p_{n-1}\subset \cdots \subset p_1\subset m$$
be the sequence of the centers of valuations $\nu_i$ with which a given valuation $\nu$ of rank $n$ is composed.\par Let
$$\Gamma\stackrel{\lambda_1}\longrightarrow \Gamma_1\stackrel{\lambda_2}\longrightarrow\cdots \stackrel{\lambda_n}\longrightarrow\Gamma_n\longrightarrow 0$$
be the corresponding sequence of morphisms of value groups. For each $\phi_{j+1}\in \nu_{j+1}(R\setminus \{0\})\subset \Gamma_{j+1}$, define $\tilde\phi_{j+1}=\hbox{\rm min}\{\nu_j(f); f\in R, \nu_{j+1}(f)=\phi_{j+1}\}$, and remark the equality of valuation ideals  $\P_{\phi_{j+1}}=\P_{\tilde\phi_{j+1}}$ with respect to the two valuations $\nu_{j+1}$ and $\nu_j$. \par
Define also $t_j=\nu_j(p_j)$ and remark that it is in the subgroup of rank one $\hbox{\rm Ker}\lambda_{j+1}$ of $\Gamma_j$, so that we may identify it with a real number. We assume chosen from the beginning an identification of all the $\hbox{\rm Ker}\lambda_j$ with subgroups of $\R$.
%----------------
\begin{theorem}\label{rk}
For each $j\in \{1,\ldots, n\}$ and every $\epsilon >0$ the sum
$$\sum_{\phi_n\in [0,t_ny_n]}\sum_{\phi_{n-1}\in [
\tilde\phi_n,\tilde\phi_n+t_{n-1}y_{n-1}]}\cdots \sum_{\phi_j\in [
\tilde\phi_{j+1},\tilde\phi_{j+1}+t_jy_j]}e_m(\P_{\phi_j}/\P^+_{\phi_j})$$
is bounded for $y_j,\ldots ,y_n$ large enough by $$(1+\epsilon)\frac{\prod_{i=j-1}^n e_{m_i}(R_{p_i}/p_{i+1}R_{p_i})}{\prod_{i=j}^n(\hbox{\rm dim}R_{p_i}/p_{i+1}R_{p_i})!} \prod_{i=j}^ny_i^{\hbox{\rm dim}R_{p_i}/p_{i+1}R_{p_i}},$$
where $m_i=p_iR_{p_i}$.
\end{theorem}
%--------------------
\par\noindent
The proof is by descending induction on $j$. The first step of the induction concerns the valuation $\nu_n$, which is of rank one and is given by considering the special case of Lemma \ref{ineq} where the valuation is of height one, and taking as additive function the multiplicity in dimension $\hbox{\rm dim}R/p_1$. Applying Proposition \ref{additivity} to the right hand side of the inequality gives:
$$\sum_{\phi\in[0,y_1t_1]}A(\P_{\phi}/\P_{\phi}^+)\leq H_{R_{p_1}}(y_1)e_m(R/p_1),$$
where $H_{R_{p_1}}(y_1)$ is the Hilbert-Samuel function of the local ring $R_{p_1}$ with respect to its maximal ideal $m_1=p_1R_{p_1}$. It is for large $y_1$ a polynomial in $y_1$ whose highest degree term is $\frac{e_{m_1}(R_{p_1})}{(\hbox{\rm dim}R_{p_1})!}y_1^{\hbox{\rm dim}R_{p_1}}$, which gives the result in this case.
%-----------
\par\medskip\noindent
Define now $ t_j=  \nu_j(p_j)\in \Gamma_j$ for each $j$ and remark that it is in the kernel of the map $\Gamma_j\to \Gamma_{j+1}$, which is of rank one. We can therefore think of it as a real number.\par\noindent 
We are going to use the following 
\begin{lemma}\label{fini} For each $j$ we have $p_j^{y_j}\P_{\phi_{j-1}}\subset \P_{\tilde\phi_{j-1}+t_jy_j}$ and the interval $[\tilde\phi_{j-1},\tilde\phi_{j-1}+t_jy_j]$ is finite.\end{lemma}\par\noindent
This follows directly from [ZS], Vol. II, appendix 3, Lemma 4.\qed\par\medskip
Assume that the result is known for $j$ and prove it for $j-1$. Let us apply the result above to \par
In view of the lemma, we have the inequality
$$\sum_{\phi_j\in [\tilde\phi_{j+1},\tilde\phi_{j+1}+t_jy_j]}e_m(\P_{\phi_j}/\P_{\phi_j}^+)\leq e_m(M_{\phi_{j+1}}/p_j^{y_j}M_{\phi_{j+1}}).$$
Now we apply Proposition \ref{additivity} and the properties of the Hilbert-Samuel polynomial of the $R_{p_j}$-module  $(M_{\phi_{j+1}})_{p_j}$, whose annihilator is $p_{j+1}R_{p_j}$,  to obtain an asymptotic estimate $$e_m(M_{\phi_{j+1}}/p_j^{y_j}M_{\phi_{j+1}})\asymp \frac{e_{m_j}((M_{\phi_{j+1}})_{p_j})}{(\hbox{\rm dim}R_{p_j}/p_{j+1}R_{p_j})!}y_j^{\hbox{\rm dim}R_{p_j}/p_{j+1}R_{p_j}},$$
so that for each $\phi_{j+1}\in \Gamma_{j+1}$ we have for any $\epsilon>0$ for $y_j$ sufficiently large (in a way which depends on $\epsilon$ and $\phi_j$) an inequality
$$\sum_{\phi_j\in [\tilde\phi_{j+1},\tilde\phi_{j+1}+t_jy_j]}e_m(\P_{\phi_j}/\P_{\phi_j}^+)\leq (1+\epsilon)\frac{e_{m_j}
\bigl((M_{\phi_{j+1}})_{p_j}\bigr)}{(\hbox{\rm dim}R_{p_j}/p_{j+1}R_{p_j})!}y_j^{\hbox{\rm dim}R_{p_j}/p_{j+1}R_{p_j}} .$$
Now we can use Corollary \ref{semic} to see that
$$e_{m_j}
\bigl((M_{\phi_{j+1}})_{p_j}\bigr)\leq e_{m}
\bigl(M_{\phi_{j+1}}\bigr)=e_m(\P_{\phi_j}/\P_{\phi_{j+1}}).$$
Summing up over the $\phi_{j+1}\in [\tilde\phi_{j+2},\tilde\phi_{j+2}+t_{j+1}y_{j+1}] $, which are finite in number, then over the $\phi_{j+2}\in [\tilde\phi_{j+3},\tilde\phi_{j+3}+t_{j+2}y_{j+2}] $ and so on, we obtain the statement.\qed\par\noindent
\textbf{Remark} If the centers of $\nu_j$ and $\nu_{j+1}$ coincide, the variable $y_j$ does not appear in the bounding expression; this corresponds to the fact that there are only finitely many elements of $\nu_j(R\setminus\{0\})$ in the union of all intervals $[\tilde\phi_{j+1},\tilde\phi_{j+1}+t_jy_j]$ (see [ZS], \textit{loc.cit.}).\par\medskip
\begin{corollary}\label{count} For each $\epsilon >0$, the number of elements of the semigroup $\nu(R\setminus\{0\})$ contained in the union of all the intervals appearing in the theorem is bounded, for $y_j$ large enough, by
$$(1+\epsilon)\frac{\prod_{i=0}^n e_{m_i}(R_{p_i}/p_{i+1}R_{p_i})}{\prod_{i=1}^n(\hbox{\rm dim}R_{p_i}/p_{i+1}R_{p_i})!} \prod_{i=1}^ny_i^{\hbox{\rm dim}R_{p_i}/p_{i+1}R_{p_i}},$$
with the convention that $p_0=m$.
\end{corollary}
  Indeed, for the valuation $\nu$ itself, the multiplicity of $\P_\phi/\P_\phi^+$ is equal to its length, because it is a finite dimensional vector space over $k=R/m$, and it is nonzero if and only if $\phi$ is in $\nu(R\setminus\{0\})$.\qed
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\bibitem{B} N. Bourbaki, Commutative Algebra, Chap. 8.
\bibitem{L} J. Lipman,
\bibitem{L-T} M. Lejeune-Jalabert and B. Teissier, {\em Normal cones and sheaves of relative jets,} Compositio Math. 28 (1974), 305--331. 
\bibitem{Te} B. Teissier, {\em Valuations, deformations, and toric geometry},  Proceedings of the Saskatoon Conference
and Workshop on valuation theory (second volume), F-V. Kuhlmann, S. Kuhlmann, M. Marshall, editors,  Fields Institute
Communications, {\bf 33}, 2003, 361-459.
\bibitem{ZS} O. Zariski, P. Samuel {\em Commutative Algebra, Vol. II},
Springer-Verlag (1960).
\end{thebibliography}


\end{document}

 