
%-------------------------------------------------
%           TEISSIER     01/05  LATEX 2e
%-----------------------------------------------
\documentclass{article}
%\usepackage{color}
\usepackage{amsthm,amsmath,amssymb,amscd}
\usepackage[greek,english]{babel}
\def \findem{\smallskip\hfill\vrule width 5 pt height 7 pt depth - 2 pt\smallskip}
%--------------------------------------------------------
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{example}[theorem]{Example}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{Remark}[theorem]{Remarks}
\renewcommand{\qedsymbol}{$\square$}
%\frenchspacing

\def\Aa{{\cal A}}
\def\A{{\mathbf A}}
\def\Pp{{\cal P}}
\def\Ll{{\mathcal L}}
\def\Qq{{\cal Q}}
\def\Q{{\mathbf Q}}
\def\Rr{{\cal R}}
\def\R{{\mathbf R}}
\def\Bb{{\cal B}}
\def\Ss{{\cal S}}
\def\C{{\mathbf C}}
\def\N{{\mathbf N}}
\def\Z{{\mathbf Z}}
\def\limproj{\mathop{\oalign{lim\cr\hidewidth$\longleftarrow$\hidewidth\cr}}}
\def\limind{\mathop{\oalign{lim\cr\hidewidth$\longrightarrow$\hidewidth\cr}}}

\begin{document}
\title
{COMPLEMENTS TO ``VALUATIONS, DEFORMATIONS, AND TORIC GEOMETRY''}

\author{Bernard Teissier} 
\maketitle

\par\medskip
\begin{center}
  PART 1:  A QUICK INTRODUCTION TO VALUATION THEORY FROM A TORIC VIEWPOINT\footnote{All references in this text are to the paper ``Valuations, deformations, and toric geometry'', Fields Institute Communications, Vol. 33, 2003, pp. 361-459, also ArXiv, Math.AC/0303200, quoted [VDTG],  and to the references of that paper.}
 \end{center}  
\par\medskip
\begin{section} {Introduction}\label{intro}
A valuation of an integral domain $R$ with field of fractions $K$ is a map $$\nu\colon K^*\to \Phi$$ with values in a totally ordered abelian
group $\Phi$ and satisfying:\par\noindent $\nu(xy)=\nu(x)+\nu(y)$,\par\noindent  $\nu(x+y)\geq\hbox{\rm
min}(\nu(x),\nu(y))$,\par\noindent $\nu(x)\geq 0$ whenever $x\in R$.\par
Since $R$ is a ring, the image $\Gamma=\nu(R\setminus\{0\})\subset \Phi_+\cup\{0\}$ is stable under addition, and so is a sub-semigroup of $\Phi_+\cup\{0\}$. If $R$ is noetherian it is well-ordered, but not finitely generated in general. According to a result of Zariski, the ordinal type of $\Gamma$ is $\omega^{h_R(\nu)}$, where $h_R(\nu)$ is an integer, the \textit{length in} $R$ of the valuation $\nu$, which is at most equal to the Krull dimension of $R$. Determining or even bounding the ordinal type of the minimal set of generators of $\Gamma$ is an open question as far as I know, although when $R$ is complete it can be shown to be at most $h_R(\nu)\omega$, so one must think of the elements $\gamma_i$ as indexed by ordinals $\leq \omega^{h_R(\nu)}$. \par Since $\Gamma$ is well ordered it has a {\it minimal} system of generators
$$\Gamma=\langle \gamma_1,\gamma_2,\ldots ,\gamma_i,\ldots \rangle ,$$ defined by an eventually transfinite construction: $\gamma_i$ is the smallest non zero element of $\Gamma$ not contained in the semigroup generated by the $(\gamma_j)_{j<i}$.\par

 \par\medskip\noindent
$\bullet$ To give a valuation of an integral domain $R$ is to choose an overring $R\subset R_\nu\subset K$ where divisibility is never a
problem:\par\noindent
Given $x,y\in R_\nu$, either $x\vert y$ or $y\vert x$ in $R_\nu$\par
Any integral domain with this property is called a \textit{valuation ring}.
Then if $U_\nu$ denotes the multiplicative group of units of $R_\nu$, we have $U_\nu\subset K^*$ and the preorder $x\leq y\iff
x^{-1} y\in R_\nu$ induces a total order of the quotient $\Phi=\frac{K^*}{U_\nu}$. The natural map $$K^*\to \Phi=\frac{K^*}{U_\nu}$$ is
a valuation. Note that the operation in $\Phi$ is noted additively although it comes from multiplication in $K^*$.\par\noindent Given a valuation $\nu$ on $K^*$, the ring $R_\nu=\{x\in K^*/\nu (x)\geq 0\}\cup\{0\}$ is a valuation ring.
\par\medskip\noindent
$\bullet$ Valuation rings appear in Geometry as 
$$\limind_{X_\alpha,\xi_\alpha}{\cal O}_{X_\alpha,\xi_\alpha}=R_\nu$$ where the limit of local rings corresponds to a projective system of points (closed or
not) in a projective system of proper birational maps of schemes (you can think of blowing-ups) which is cofinal in the system of all proper birational maps. A {\it sequence} of points in such a system looks like:
$$\begin{array}{lr}
\cdots X_i\rightarrow X_{i-1}\rightarrow\cdots\rightarrow X_1\rightarrow X_0=X\cr
\ldots\  \xi_i\rightarrow \ \xi_{i-1}\rightarrow\cdots\rightarrow\ \xi_1\rightarrow \ \xi_0=\xi\end{array}$$
$\bullet$ If there was a cofinal system in the system of ${\cal O}_{X_\alpha,\xi_\alpha}$ consisting of {\it regular} local rings, then $R_\nu$, which is ``regular'' but very large, in particular not noetherian in general, could be approximated by regular local rings.
\par\medskip\noindent
 This is the problem of {\it local uniformization}, solved by Zariski in 1940 for algebraic varieties over a field $k$ of
characteristic zero as a step towards resolution of singularities, and which is also a consequence of Hironaka's resolution theorem.
For positive characteristic, in dimension $\leq 3$, it is a consequence of the results of Abhyankar . \par\medskip\noindent
 $\bullet$
  Given a field extension $k\to K$, the restriction to $k$ of a valuation of $K$ is a valuation of $k$. The valuations of $K$ which restrict to the trivial valuation with value group $0$ on $k$ are known as valuations of $K/k$. The traditional view is to classify them according to the ``size'' of their ordered group $\Phi$ of values, and the ``size'' of the extension $k\to k_\nu=R_\nu/m_\nu$. The ``size'' of $k_\nu$ relative to $k$ is measured by the transcendence degree of this field extension. The ``size'' of $\Phi$ is measured in two ways: the \textit{rational rank} $\hbox{\rm r}(\Phi)=\hbox{\rm dim}_{\Q}\Phi\otimes_{\Z}\Q$ and the \textit{rank} of $\Phi$, which I prefer to call the height, and which is the length of a maximal chain of convex subgroups of the ordered group $\Phi$. It is also the Krull dimension of the ring $R_\nu$.
A classical problem going back to Krull is to see how the valuation ring of a valuation of $K/k$ differs from the ring $k[[t^{\Phi_+}]]$ of formal series with \textit{well ordered} sets of exponents in $\Phi_+$, equipped with the valuation associating to each series the smallest exponent appearing in it.\par\medskip\noindent
$\bullet$
 We are interested here in pairs $(R,\nu)$, where $R$ is a noetherian ring and $\nu$ a valuation of its field of fractions, non negative on $R$. In addition to the measures of the complexity of the valuation just mentionned, we need to consider the values of $\nu$ on $R\setminus \{0\}$. These values form a subsemigroup $\Gamma$ of the semigroup $\Phi_+\cup\{0\}$. Since $R$ is noetherian, the set $\Gamma$ is \textit{well ordered}, which is not the case for $\Phi_+\cup\{0\}$ in general. This implies that the semigroup $\Gamma$ has a unique minimal set of generators. The group $\Phi$ being given, this set of generators contains some information about the singularities of $R$. \par
 It is therefore interesting to try to build noetherian rings with a valuation having a semigroup of values given in advance. This is the content of the next section.
\end{section}
\begin{section}{Examples}\label{examp}
 Let $\Gamma$ be a sub semigroup of $\Q_+$. We start by building integral domains with a valuation $\nu$ having value group
$\Phi\subseteq\Q$ and such that the semigroup of the values taken by the valuation on $R$ is exactly $\Gamma$. 
\par\medskip\noindent
\textbf{Example 1}: $\Gamma=\langle \gamma_1,\ldots ,\gamma_{g+1}\rangle\subset \N$ where $\langle A\rangle$ means the semigroup of all non negative integral linear combinations of the elements of $A$ and the $\gamma_i$ are collectively coprime integers, ordered according to their indices. We assume that they form a minimal system of generators of $\Gamma$, which means that for all $i$, we have $\gamma_{i+1}\notin\langle \gamma_i,\ldots ,\gamma_i\rangle$.\par\medskip\noindent\textbf{Example 2}: Let $(s_i)_{i\geq 1}$ be a sequence of positive integers such that $s_i\geq 2$ for $i\geq 2$, and define inductively rational numbers $(\gamma_i)_{i\geq 1}$ by the relations
$$\gamma_1=\frac{1}{s_1}\ ,\ \ \ \gamma_{i+1}=s_i\gamma_i+\frac{1}{s_1\ldots s_{i+1}}$$
Then take $$\Gamma=\langle \gamma_1,\gamma_2,\dots ,\gamma_i,\ldots\rangle\subset \Q_+.$$ 
If $\forall i\ s_i=i $, the group $\Phi$ generated by $\Gamma$ is $\Q$.
\par\medskip\noindent
To such semigroups we can associate geometric objects, the spectra of their semigroup algebras over an algebraically closed field $k$. The Krull dimension  of the semigroup algebra is equal to the rational rank $\hbox{\rm dim}_\Q \Phi\otimes_\Z\Q$ of the group $\Phi$ generated by $\Gamma$.\par In our case this rank is equal to one, so that our semigroup algebras correspond to curves. In example 1, we have a  monomial curve in $\A^{g+1}(k)$: $$u_i=t^{\gamma_i},\ 1\leq i\leq g+1,\ \ \gamma_i\in \N.$$\par\noindent
Example 2 must also be thought of as a monomial curve $u_i=t^{\gamma_i}$ but now the embedding dimension is infinite.\par\medskip\noindent
We are now going to deform these rings, and for that we need equations for them. These equations correspond to relations with integral coefficients between the generators $\gamma_i$.
The equations defining the monomial curve of example 1, that is,  the relations between the $\gamma_i$, may be fairly complicated. We shall make the following simplifying assumptions:\par\noindent
a) if $e_i$ is the gcd of $(\gamma_1,\ldots, \gamma_i)$ and if we write $e_i=s_{i+1}e_{i+1}$, then for $1\leq i\leq g$
$$s_{i+1}\gamma_{i+1}\in \langle\gamma_1,\ldots ,\gamma_i\rangle$$\par\noindent
b) $s_i\gamma_i<\gamma_{i+1}$ for $2\leq i\leq g$.\par
Then the relations are generated by the following $g$ expressions of condition a):\par\noindent
$$s_{i+1}\gamma_{i+1}=\sum_{k=1}^i\ell^{(i+1)}_k\gamma_k,\ \hbox{\rm with }\ell^{(i+1)}_k\in \N$$
\par\medskip\noindent
These relations are not uniquely determined but in view of condition a) there is a unique way of writing each relation satisfying the condition that $\ell^{(i+1)}_k< s_k$ for $2\leq k\leq i$. Condition a) implies that in the special case considered the monomial curve is a complete intersection with equations 
$$u_{i+1}^{s_{i+1}}-\Pi_{k=1}^{i}u_k^{\ell^{(i+1)}_k}=0,\ 1\leq i\leq g.$$
In example 2, it is not difficult to see, using the fact that $(\gamma_1,\ldots, \gamma_i)$ are in the subgroup of $\Q$ consisting of rational numbers which can be written with denominator $s_1\ldots s_i$, that all relations are generated by the:
$$s_{i+1}\gamma_{i+1}=\sum_{k=1}^i\ell^{(i+1)}_k\gamma_k\ \hbox{\rm with }\ell^{(i+1)}_k\in \N , i\geq 1,$$ so that the equations of our monomial curve are
$$u_{i+1}^{s_{i+1}}-\Pi_{k=1}^iu_k^{\ell^{(i+1)}_k}=0,\ 1\leq i.$$
\par\medskip\noindent
All these equations are binomial equations defining irreducible varieties in a possibly infinite dimensional affine space. They are (non normal) toric varieties. \par\medskip
  Let us now remark that in both examples we have $\gamma_{i+2}>s_{i+1}\gamma_{i+1}$ and let us deform the equations in the following manner: in the first example we consider a variable $v$ and the equations 
 $$\begin{array}{lr}u_{i+1}^{s_{i+1}}-\Pi_{k=1}^{i}u_k^{\ell^{(i+1)}_k}-vu_{i+2}&=0,\ 1\leq i\leq g-1.\\
 u_{g+1}^{s_{g+1}}-\Pi_{k=1}^{g}u_k^{\ell^{(g+1)}_k}&=0\end{array}$$
In the second example, we introduce a variable $v_i$ for each index $i\geq 2$ and consider for $i\geq 1$ the equations
$$u_{i+1}^{s_{i+1}}-\Pi_{k=1}^iu_k^{\ell^{(i+1)}_k}-v_{i+1}u_{i+2}=0$$
\par\medskip\noindent
In both cases we have an obvious elimination process in the polynomial ring $k[(v_j^{\pm 1});(u_i)]$. 
Setting $v_j=1$, the results are isomorphisms:\par\medskip\noindent
$\bullet$ For example 1:
\small{
$$\begin{array} {lr} R=k[u_1,u_2]/(F)& \\ =k[u_1,\ldots ,u_{g+1}]/(u_2^{s_2}-u_1^{\ell^{(2)}_1}-u_3, u_3^{s_3}-u_1^{\ell^{(3)}_1}u_2^{\ell^{(3)}_2}-u_4,\ldots ),\end{array}$$
}
where $F(u_1,u_2)$ is the result of the elimination; for example if $\Gamma=\langle 4,6,13\rangle$, the equations of the monomial curve are 
$$\begin{array}{lr}u_2^2-u_1^3=&0\\
u_3^2-u_1^5u_2=&0\end{array}$$
and since the deformation affects only the first equation and is $u_2^2-u_1^3-vu_3=0$, we find 
$$F(u_1,u_2)=(u_2^2-u_1^3)^2-u_1^5u_2=0.$$
\par\medskip\noindent
$\bullet$ For example 2:
\small{
$$\begin{array} {lr} R=k[u_1,u_2]& \\ =k[u_1,\ldots ,u_i,\ldots ]/(u_2^{s_2}-u_1^{\ell^{(2)}_1}-u_3, u_3^{s_3}-u_1^{\ell^{(3)}_1}u_2^{\ell^{(3)}_2}-u_4,\ldots )\end{array}$$
}
In both cases, giving to the variable $u_i$ the weight $\gamma_i$ determines a valuation on the ring and the isomorphism gives a way to compute it: \textit{ in the right hand side the value of a polynomial $P(u_1,u_2)$ rewritten replacing systematically each $u_i^{s_i}$ by $\Pi_{k<i}u_k^{\ell^{(i)}_k}+u_{k+1}$  is the minimum of the values (i.e., weights) of its monomials} and this determines a valuation of $R$ with semigroup $\Gamma$.
\par\medskip\noindent
Giving to the variable $u_i$ the weight $\gamma_i$ determines a monomial order on the polynomial ring $k[(u_i)]$, and therefore a filtration by the minimal order of the monomials in a polynomial.
Each of the equations which we have created by deformation has an initial form with respect to this filtration which is precisely the binomial equation which we have deformed.\par
In fact we have a faithfully flat family parametrized by $k[(v_j)]$ specializing the ring $R$ to the ring of the monomial curve defined by the initial binomial equations. The equations of $R$ play the role of a \textit{standard, or Gr\"obner, basis} with respect to the monomial order.\par
Since we have seen that the Krull dimension of that ring is equal to one, the second example contradicts the semicontinuity of fiber dimensions in a family. There is no absurdity since that semicontinuity is proved under a finiteness condition which is not fulfilled here. \par The situation in our case can be described by what I call \textit{the abyssal phenomenon}:
\par\medskip\noindent
Let ut write out the system of equations appearing in the second example:
$$\begin{array}{lr}
u_2^{s_2}-u_1^{\ell^{(2)}_1}-v_2u_3&=0\cr
u_3^{s_3}-u_1^{\ell^{(3)}_1}u_2^{\ell^{(3)}_2}-v_3u_4&=0\cr
\vdots &\vdots\cr
u_{i+1}^{s_{i+1}}-\Pi_{k=1}^iu_k^{\ell^{(i+1)}_k}-v_{i+1}u_{i+2}&=0\cr
\vdots &\vdots
\end{array}
$$
When all $v_i$ are nonzero, say equal to one, it amounts actually to an endless sequence of substitutions and therefore it cannot decrease the dimension, while when we specialize to the monomial curve, making all $v_{i+1}=0$, we obtain equations which also express all $u_i,\ i\geq 3$ algebraically in terms of $(u_1,u_2)$, but now $u_1$ and $u_2$ are algebraically dependent so that the dimension drops to 1.
\par\medskip\noindent
 However, if $k$ is of characteristic zero we can also view these equations as defining a very transcendental curve in $\A^2(k)$ whose Zariski closure, which is all our equations see, is the entire affine plane.\par

To see this, use the order on the $\gamma_j$ to order the equations $u_j^{s_j}-\ldots$ as we did above, and for a given $n\in \N$ truncate the system at order $n$ in the following sense: keep all equations of index $<n$, replace the equation of index $n$ by its initial form, which involves only variables of index $\leq n$, and forget all the other equations. We are now reduced to the case of example 1 except that we have to multiply $\gamma_1,\ldots ,\gamma_n$ by their common denominator, obtaining a sequence of coprime integers.
\par\medskip\noindent
If $k$ is of characteristic zero, we can solve the corresponding equation $F_n(u_1,u_2)=0$ by a Puiseux expansion
$$u^{(n)}_2=\sum_{j=1}^\infty a^{(n)}_ju_1^{\frac{j}{s_2\ldots s_n}}$$
 Using the Smith-Zariski formula for the intersection number of parametrized curves one can show that as $n$ increases these Puiseux expansions converge in the ring $k[[t^{\Q_+}]]$ of series with well ordered sets of rational exponents to a series $w(u_1)$ of fractional powers of $u_1$ \textit{whose exponents have unbounded denominators}. Because of Newton-Puiseux, setting $u_2=w(u_1)$  cannot cause the vanishing of a polynomial (or even a power series) in $(u_1,u_2)$. It parametrizes a transcendental curve whose Zariski closure is $\A^2(k)$.
 \par\medskip\noindent
We have produced it as a deformation of an algebraic, indeed toric, curve of infinite embedding dimension.\par
This curve defines a valuation on $k[u_1,u_2]$ by taking, for any polynomial $P(u_1,u_2)$, the order in $u_1$ of $P(u_1,w(u_1))$, which is finite as we just saw. It is of course the same valuation as that which is obtained as explained above.\par
\par\medskip\noindent
Remark that we have many choices to deform our monomial curve of infinite embedding dimension:\par\noindent
Given an integer $t\geq 3$, we could deform by
$$\begin{array}{lr}
u_2^{s_2}-u_1^{\ell^{(2)}_1}-v_2u_t&=0\cr
u_3^{s_3}-u_1^{\ell^{(3)}_1}u_2^{\ell^{(3)}_2}-v_3u_{t+1}&=0\cr
\vdots &\vdots\cr
u_t^{s_t}-u_1^{\ell^{(t)}_1}...u_{k-1}^{\ell^{(t)}_{t-1}}-v_tu_{2t-2}&=0\cr
\vdots &\vdots\cr
u_{i+1}^{s_{i+1}}-\Pi_{k=1}^iu_k^{\ell^{(i+1)}_k}-v_{i+1}u_{i+t-1}&=0\cr
\vdots &\vdots
\end{array}
$$
And then our deformed ring is $k[u_1,\ldots ,u_{t-1}]$, with a valuation which still has the same semigroup $\Gamma$.
\par\medskip\noindent
We can also deform our curve into a singular space: fix an integer $t\geq 2$ and for $j=2,\ldots, t$ a polynomial $P_j(u_1,\ldots ,u_t)$ in which every monomial has weight $>s_j\gamma_j$ and which contains no linear term. Fix another integer $t'\geq 2$ and then consider the deformation
$$\begin{array}{lr}
u_2^{s_2}-u_1^{\ell^{(2)}_1}-v_2P_2(u_1,\ldots ,u_t)&=0\cr
u_3^{s_3}-u_1^{\ell^{(3)}_1}u_2^{\ell^{(3)}_2}-v_3P_3(u_1,\ldots ,u_t)&=0\cr
\vdots &\vdots\cr
u_t^{s_t}-u_1^{\ell^{(t)}_1}...u_{t-1}^{\ell^{(t)}_{t-1}}-v_tP_t(u_1,\ldots ,u_t)&=0\cr
u_{t+1}^{s_{t+1}}-u_1^{\ell^{(t+1)}_1}...u_{t}^{\ell^{(t+1)}_t}-v_{t+1}u_{t+t'}&=0\cr
\vdots &\vdots\cr
u_{i+1}^{s_{i+1}}-\Pi_{k=1}^iu_k^{\ell^{(i+1)}_k}-v_{i+1}u_{i+t'}&=0\cr
\vdots &\vdots
\end{array}
$$
Now taking all $v_s=1$, we obtain a {\it singular} ring, with a valuation still having semigroup $\Gamma$. If we take infinitely many $P_j$, the ring is no longer noetherian.
\end{section}
\begin{section}{Relation with local uniformization}\label{localun}
  Now what is the use of this? \par
  Suppose that we are interested in building a \textit{regular} local ring $R'$ between $R$ and $R_\nu$, which is essentially of finite type over $R$; this is the problem of local uniformization. 
 The basic idea is that over an algebraically closed field of any characteristic it is not difficult to resolve the singularities of an irreducible variety defined by binomials, by toric maps. In the case of the first example, that toric resolution will also resolve the singularities of the plane curve obtained by elimination, viewed as embedded in affine $g+1$-space; this is due to the fact that the deformation adds only terms of higher weight than the initial binomial equation.\par
 The second example is not so convincing as far as local uniformization is concerned since the ring is regular, and also because there is no resolution in the usual sense of a space defined by infinitely many binomial equations. 
\par\medskip\noindent
 But as we saw above, we can make it more complicated as follows: in some of the equations instead of adding a linear term $u_{j+2}$, we can add with a factor $v_{i+2}$ a polynomial, or even a series, where every monomial is of weight greater that the weight of the initial binomial and where there is no linear term, and thus manufacture a singular ring, which may be of dimension $>2$. When we grow tired, we go back to adding linear terms as above, which we must do anyway if we want the result to be noetherian. The resulting ring still specializes to the ring of the monomial curve wich admits as equations the initial forms, and therefore a toric resolution of the binomial variety corresponding the finitely many binomials which we have deformed in this new manner will provide a local uniformization of the valuation defined on $R$. \par\medskip\noindent    In short, if we have manufactured a complicated singular ring $R$ by adding non linear terms of higher weight to a finite number of the binomial equations of our curve, we may be glad to exchange its noetherianity for the simplicity of dealing with a toric variety, provided we can show that it suffices to resolve {\bf finitely many} of the binomials to uniformize the valuation on $R$.\par
 \par\medskip\noindent
 Now the claim is that {\bf this is essentially the general situation}, at least when $R$ is a complete equicharacteristic noetherian local ring with an algebraically closed residue field.
 \par\medskip\noindent
Let $R\subset R_\nu$ be the inclusion of a ring in a valuation ring. The only really important case for local uniformization is when $R$ is an excellent equicharacteristic local ring and $R_\nu$ dominates $R$, i.e., $m_\nu\cap R=m$, and the residue field extension $R/m\subset R_\nu/m_\nu$ is trivial; we say then that $\nu$ is a \textit{rational} valuation of $R$.\par\noindent
 We may consider the filtration of $R$ by the ideals 
 $$\begin{array}{lr}\Pp_\phi(R)=\{x\in R/\nu(x)\geq\phi\}\\
 \Pp^+_\phi(R)=\{x\in R/\nu(x)>\phi\}\end{array}$$ and this gives us an associated graded ring
 $$\hbox{\rm gr}_\nu R=\bigoplus_{\phi\in \Gamma}\frac{\Pp_\phi(R)}{\Pp^+_\phi(R)},$$
 where $\Gamma\subset \Phi_+\cup\{0\}$ is the semigroup of the values taken on elements of $R\setminus\{0\}$ by the valuation $\nu$ with value group $\Phi$. Remark that if $\phi\notin\Gamma$, then $\Pp_\phi(R)=\Pp^+_\phi(R)$ and that if the valuation $\nu$ is rational and $\phi\in \Gamma$, then the quotient is a one-dimensional vector space over $k=k_\nu$.\par
  The first basic fact is
 \par\medskip\noindent
 \textbf{Proposition 1:}
 \textit{If $\nu$ is a rational valuation of the noetherian ring local $R$, the associated graded ring can be presented as a quotient of a polynomial ring in countably many variables by a binomial ideal}:
 $$\hbox{\rm gr}_\nu R=k[(U_i)_{i\in I}]/(U^m-\lambda_{mn}U^n)_{m,n\in E}.$$
 Now one can associate to $(R,\nu)$ a \textit{valuation algebra}
 $$\Aa_\nu(R)=\bigoplus_{\phi\in \Phi}\Pp_\phi(R)v^{-\phi}\subset R[v^\Phi],$$ where $R[v^\Phi]$ is the group algebra of $\Phi$ with coefficients in $R$.\par
 If $R$ contains a field $k$ such that the valuation takes the value $0$ on $k^*$, we have a natural composed map
 $$k[v^{\Phi_+}]\to R[v^{\Phi_+}]\to 
 \Aa_\nu (R),$$ corresponding to a map of schemes 
 $$\hbox{\rm Spec}\Aa_\nu(R)\to\hbox{\rm Spec}k[v^{\Phi_+}].$$
 \par\medskip\noindent And the second basic fact is:\par\noindent
  \textbf{Proposition 2:}
 \textit{If in addition the ring $R$ contains a field of representatives, the $k[v^{\Phi_+}]$-algebra $\Aa_\nu(R)$ is faithfully flat, the general fiber of the corresponding map of schemes is isomorphic to $\hbox{\rm Spec}R$ and its special fiber is $\hbox{\rm Specgr}_\nu R$.}\par
 By a result of Piltant, the Krull dimension of $\hbox{\rm Specgr}_\nu R$ is the rational rank of the group $\Phi$ of the valuation, and by Abhyankar's inequality, we have for a rational valuation
 $$\hbox{\rm dimgr}_\nu R\leq \hbox{\rm dim} R.$$
 Strict inequality can occur, as we saw in example 2.
 \par\medskip\noindent
 Using the properties of flatness, one can deduce from this a valuative version of Cohen's theorem:\par\noindent
\begin{theorem}\label{cohen} If the noetherian local ring $R$ is complete and equicharacteristic, given a field of representatives $k\subset R$ and elements $\xi_i\in R$ whose initial forms $\overline\xi_i$ generate the $k$-algebra $\hbox{\rm gr}_\nu R$, the surjective map of $k$-algebras
 $$(*)\ \ \ \ \ \ \ \ \ \ \ \ \ \ k[(U_i)_{i\in I}]\to\hbox{\rm gr}_\nu R\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$
 of Proposition 1, mapping $U_i$ to $\overline\xi_i$, extends to a continuous surjective map of $k$-algebras
 $$(**)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \widehat{k[(u_i)_{i\in I}]}\to R\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$
 mapping $u_i$ to $\xi_i\in R$, and 
 such that the associated graded map with respect to the natural filtrations coincides with the map (*)
 \end{theorem}
 \par
 Here the hat means a \textit{scalewise} completion taking into account the structure of the group $\Phi$ of the valuation.
 \par\medskip\noindent
 Moreover the kernel of the map (**) is generated, up to closure, by equations which are deformations of the binomial equations $U^m-\lambda_{mn}U^n$ generating the kernel of the map (*). They are of the form
 $$F_{mn}=u^m-\lambda_{mn}u^n+\sum_pc_pu^p,$$ with $ c_p\in k^*,\ w(u^p)>w(u^m)=w(u^n)$, and $w$ is the monomial weight giving to $u_i$ the weight $\gamma_i=\nu (\xi_i)\in \Gamma$. \par
 Now because $R$ is noetherian, its maximal ideal is generated by finitely many of the $\xi_i$ and any variable $u_j$ not in that finite set must appear \textit{linearly} in one of the equations $F_{mn}$; In fact, modulo an implicit function theorem whose proof is not yet written in full,  one can prove that all the equations $F_{mn}$ except finitely many must be of the form
 $$F_i=u^{n(i)}-\lambda_iu^{m(i)}+c_iu_i+\sum_pc^{(i)}_pu^p,\ \ \ c_i\in k^*$$
\par\medskip\noindent 
This is beginning to look a lot like our example 2. 
It suffices to resolve the toric variety defined by the finitely many binomials which do not appear in the $F_i$, this will extend to a birational toric map $Z\to {\rm Spec}R$ such that $Z$ is non singular at the point picked by the valuation $\nu$, and all the $F_i$ add nothing: it is just a graph.\par
This reduces us to the case where the graded $k$-algebra $\hbox{\rm gr}_\nu R$ is finitely generated: one proves that an irreducible binomial variety over an algebraically closed field of any characteristic has  embedded resolutions by  toric maps (joint work with P. Gonz\'alez P\'erez), and then that the same toric map also resolves the space defined by the deformed equations at the point picked by the valuation provided one deforms by adding terms of higher weight.  More precisely:
\end{section}
\begin{section}{Overweight deformations of binomial ideals}
Let us go back to the situation of a rational valuation on a local noetherian integral domain $R$ with semigroup $\Gamma=\langle \gamma_1,\gamma_2, \ldots\rangle$. Recall ([VDTG], Proposition 4.15) that the graded algebra $\hbox{\rm gr}_\nu R_\nu$ of a valuation ring is the union of a nested sequence of polynomial algebras in $N=\hbox{\rm r}(\nu)$ variables over $k_\nu=R_\nu/m_\nu$ with maps between them sending each variable to a term. These subalgebras are essentially the semigroup algebras of nested free subsemigroups of the non negative part $\Phi_+\cup\{0\}$ of the value group $\Phi$ of the valuation.\par

Let $\N^N\subset \Phi_+\cup\{0\}$ be such a free subsemigroup of the non negative part of a totally ordered group $\Phi$. Let $(\gamma_i)_{i\in F}$, where $F=\{1,\ldots, f\}$ be the elements of the sequence $(\gamma_i)_{i\geq 1}$ which are in $\N^N$. They form a minimal set of generators of a semigroup $\Gamma_F\subseteq \Gamma\bigcap\N^N$. Given any field $k$ this determines a \textit{monomial order} on $k[(U_i)_{i\in F}]$ corresponding to the order on the weights $w(U_i)=\gamma_i$ given by the order of $\Phi$  and a map of semigroup algebras over $k$
$$k[(U_i)_{i\in F}]\to k[t_1,\ldots ,t_N].$$
The kernel of this map is a prime ideal $I$ of $k[(U_i)_{i\in F}]$ generated by binomials $u^{m^\ell}-\lambda_\ell u^{n^\ell}$. These binomials are such that the vectors $m^\ell-n^\ell$ generate the lattice of relations between the elements $(\gamma_j)_{j\in F}$, and the constants $\lambda_\ell\in k^*$ are determined by the valuation on $R$. \par
Let $w$ be a weight on a polynomial or power series ring over a field $k$, with values in the positive part $\Phi_+$ of a totally ordered group $\Phi$ of finite rational rank. \par Let us consider the power series case and the ring $S=k[[u_1,\ldots ,u_N]]$. Consider the filtration of $S$ indexed by $\Phi_+$ and determined by the ideals $\Qq_\phi$ of elements of weight $\geq \phi$, where the weight of series is the minimum weight of a monomial appearing in the series. Defining similarly $\Qq^+_\phi$ the ideal of elements of weight $>\phi$, the graded ring associated to this filtration is $$
\bigoplus_{\phi\in \Phi^+\cup\{0\}}\frac{\Qq_\phi}{\Qq^+_\phi}=k[U_1,\ldots ,U_N].$$
\begin{definition}\label{overweight}
Given a weight $w$ as above, a (finite dimensional) valuative overweight deformation is the datum of a prime binomial ideal $(u^{m^\ell}-\lambda_\ell u^{n^\ell})_{1\leq \ell \leq s}$ such that the vectors $m^\ell-n^\ell$ generate the lattice of relations between the $\gamma_i=w(u_i)$ and of series 
$$\begin{array}{lr}
F_1=u^{m^1}-\lambda_1 u^{n^1}+\Sigma_{w(p)>w(m^1)} c^{(1)}_pu^p\\
F_2=u^{m^2}-\lambda_2 u^{n^2}+\Sigma_{w(p)>w(m^2)} c^{(2)}_pu^p\\
.....\\
F_\ell=u^{m^\ell}-\lambda_\ell u^{n^\ell}+\Sigma_{w(p)>w(m^\ell)} c^{(\ell)}_pu^p\\
.....\\
F_s=u^{m^s}-\lambda_s u^{n^s}+\Sigma_{w(p)>w(m^s)} c^{(s)}_pu^p\\
\end{array}\leqno{(OD)}
$$ in $k[[u_1,\ldots ,u_N]]$ 
such that, with respect to the monomial order determined by $w$, they form a Gr\"obner basis for the ideal which they generate.\par\noindent 
Here we have written $w(p)$ for $w(u^p)$ and the coefficients $c^{(\ell)}$ are in $k$.
\end{definition}
Let us agree to call $X$ the formal subspace of $\A^N(k)$ defined by te equations $F_\ell$.
\begin{proposition}\label{resoverwght}\par\noindent
a) A valuative overweight deformation determines a valuation $\nu$ of the ring $R=S/(F_1,\ldots ,F_s)$, and the associated graded ring of $R$ is $$\hbox{\rm gr}_\nu R=k[U_1,\ldots ,U_N]/(U^{m^1}-\lambda_1 U^{n^1},\ldots ,U^{m^s}-\lambda_s U^{n^s}).$$\noindent
b) There exists a birational toric map $Z\to \A^N(k)$ such that the strict transform of 
$X$ is regular at the point picked by $\nu$.
\end{proposition}
\begin{proof} To prove a), let us first consider the group of rank one $\Phi_1$ which is a quotient of $\Phi$ by its largest non trivial convex subgroup. By composition the weight $w$ gives rise to a weight $w_1$ with values in $\Phi_1$. Let us denote by $p_1$ the ideal of elements of $S$ whose $w_1$ weight is $>0$. If the preimages in $S$ of an element $x$ have unbounded weights, there is a sequence of such preimages whose $w_1$ weights tend to infinity. Since the intersection $\bigcap_{\phi_1\in \Phi_1^+}\Qq_\phi$ is zero and $S$ is complete, by Chevalley's theorem any preimage of $x$ then lies in $\bigcap_N(I+ M^N)$ where $M=(U_1,\ldots ,U_N)$ and $I=(F_1,\ldots ,F_s)$. This intersection is equal to $I$ by Krull's theorem so that $x=0$.\par Therefore the inverse images in $S$ of a non zero element of $R$ have bounded weight. Again by noetherianity of $S$ this implies that those weights have a maximum in $\Phi_+$.\par
We can therefore define the order of a non zero element of $R$ as the maximum weight of a series of $S$ having $x$ as image under the quotient map $S\to R=R/I$. This clearly satisfies the inequalities $\nu (x+y)\geq \hbox{\rm min}(\nu(x),\nu(y))$ and $\nu(xy)\geq \nu(x)+\nu(y)$.  Thus, we have defined an order function on $R$. To prove that it is a valuation is to prove that the second inequality is an equality, and we argue as follows:\par\noindent
The order function $\nu$ determines a filtration on $R$; by construction the associated graded ring of $R$ with respect to this filtration is a quotient of the associated graded ring $k[U_1,\ldots ,U_N]$ of $S$ with respect to the weight filtration. Indeed if we denote by $\cal Q_\phi$ the ideal of elements of weight $\geq \phi$ in $S$, we see that by definition of $\nu$ it maps onto the ideal of elements of $R$ which are of order $\geq \phi$. \par
The ideal defining the quotient is the ideal of $k[U_1,\ldots ,U_r]$ generated by the initial forms of the elements of $I$ with respect to the weight filtration of $S$. Since by hypothesis this initial ideal is generated by the $U^{m^\ell}-\lambda_\ell U^{n^\ell}$, the graded algebra $\hbox{\rm gr}_\nu R$ is equal to $k[U_1,\ldots ,U_N]/I_0$ where $I_0$ is the ideal generated by the binomials. It is therefore an integral domain, which shows that the order function $\nu$ is actually a valuation.
\par\medskip\noindent
To prove b), consider the system of hyperplanes $H_\ell =H_{m^\ell-n^\ell}$ of $\check \R^N$ dual to the vectors $m^\ell-n^\ell$ and remember from [VDTG] (or see below) that if $\Sigma$ is a regular fan subdividing the first quadrant of $\check \R^N$ and compatible with all the the $H_\ell$, then the toric modification $\pi(\Sigma)\colon Z\to \A^N$ of $\A^N$ determined by $\Sigma$ is an embedded resolution of singularities of the toric variety $X_0$ corresponding to the binomial ideal. Remember also that the charts of $Z$ where the strict transform of $X_0$ meets the exceptional divisor are those which correspond to cones $\sigma=\langle a^1,\ldots ,a^N\rangle$ which meet the \textit{weight cone} which is the intersection of the $d$ dimensional vector space $\bigcap_{\ell=1}^tH_\ell$ with the first quadrant.\par 
More precisely:\par
To the regular fan $\Sigma$ with support $\R_{\geq 0}^N$ corresponds a proper and birational toric map of non singular toric varieties $\pi(\Sigma)\colon Z(\Sigma)\to \A^N(k)$ . To each cone of maximal dimension $\sigma=\langle a^1,\ldots , a^N \rangle$ corresponds a chart $Z(\sigma)$ of $Z(\Sigma)$ which is isomorphic to $\A^N(k)$. If we choose adapted coordinates $y_1,\ldots ,y_N$ in that chart, the restriction
$$\pi (\sigma)\colon Z(\sigma)\to \A^N(k)$$ is described by monomials:\par\noindent
$$\begin{array}{lr}
 u_1=& y_1^{a_1^1}\ldots y_N^{a_1^N}\\
 u_2=& y_1^{a_2^1}\ldots y_N^{a_2^N}\\
 ...\ \ \ \ \ \ \\
 u_N=& y_1^{a_N^1}\ldots y_N^{a_N^N}\\,
 \end{array}\leqno{(*)}$$
 and so for each monomial $u^m$ we have
 $$u^m\mapsto
y_1^{\langle a^1,m\rangle}\cdots y_N^{\langle a^N,m\rangle},$$ where $\langle a^i,m\rangle =\sum_{j=1}^Na^i_jm_j$. \par\noindent Let us
compute the transform by this map of one of our binomial generators, denoted by $u^m-\lambda u^n$. We may assume that
$a^1,\ldots ,a^t$ are those among the $a^j$ which lie on the hyperplane $H_{m-n}$ dual to
$m-n$,  i.e., such that $ \langle
a^j,m-n\rangle=0\ ,\ 1\leq j\leq t$. \textit{Because our fan is compatible with $H_{m-n}$,  all the other $\langle a^j,m-n\rangle$ are of the same sign}, say
$\langle a^j,m-n\rangle>0$. We have then
$$u^m-\lambda u^n\mapsto y_1^{\langle a^1,n\rangle} \cdots y_N^{\langle a^N,n\rangle}(y_{t+1}^{
\langle
a^{t+1},m-n\rangle}\cdots y_N^{\langle a^N,m-n\rangle}-\lambda).$$ The strict transform $y_{t+1}^{
\langle
a^{t+1},m-n\rangle}\cdots y_N^{\langle a^N,m-n\rangle}-\lambda=0$ of our binomial hypersurface is non singular, and transversal to the exceptional divisor defined by $
\Pi_{j\in J}y_j=0$, where $J$ is the set of those $j,1\leq j\leq N$ such that $a^j$ is not a basis vector.\par\noindent
It is shown in [VDTG] that the strict transform by the map $\pi(\Sigma)$ of the toric variety $X\subset \A^N(k)$ defined by our prime binomial ideal is non singular and transversal to the exceptional divisor.\par\noindent
Furthermore, if the intersection of $\sigma$ with $W$ is $d$-dimensional, then the equations of the strict transform depend on exactly $d$ variables, say $y_{t+1},\ldots ,y_N$ and in view of their binomial nature their only solutions in the charts $Z(\sigma$ are given by $y_{t+j}=c_{t+j}\in k^*$.
\par\medskip
We are going to show that there exist regular fans refining such a fan $\Sigma$ such that the corresponding toric modification resolves the strict transform of $X$ at the point picked by the valuation $\nu$.\par
 Let us examine how valuative overweight deformations behave with respect to toric resolutions of singularities. 
 Let $w$ be a weight on the variables $u_1,\ldots , u_N$ with values in a well ordered subsemigroup of the positive semigroup $\Phi_+$ of a totally ordered group $\Phi$ of finite height. Let us say that a convex rational cone $\sigma\subset \check\R^N$ is $w$-\textit{centering} if the monomial valuation on $k(u_1,\ldots , u_N)$ determined by $w$ has a center in $k[\check\sigma\cap\Z^N]$. If we assume that $\sigma$ is simplicial and write as above $\sigma=\langle a^1,\ldots ,a^N\rangle$, this means that in the map determined by $\sigma$, we have that the $w(y_i)$ are $\geq 0$. \par
 We remark that since the matrix of the $a^j_1$ is invertible, the weights of the $y_j$ are uniquely determined by the $w(u_i)$.\par\noindent
 If the weight $w$ is of rank one and we identify its value group $\Phi$ with a subgroup of $\R$, we can consider the vector ${\mathbf w}=(w(u_1),\ldots ,w(u_N))\in \R_{\geq 0}^N$. Then the positivity of the $w(y_j)$ is equivalent to the fact that ${\mathbf w}$ is in $\sigma$; a simplicial convex cone $\sigma$ is $w$-centering if and only if it contains the vector ${\mathbf w}$.\par If the rank ${\rm r}$ of $\Phi$ is greater than one, we consider the sequence of convex subgroups, with the convention that $\Psi_0=\Phi$: 
 
 $$(0)=\Psi_h\subset \Psi_{h-1}\subset\ldots \Psi_1\subset \Phi,$$ and we notice that since $\Phi$ has no torsion and we are interested only in inequalities and equalities we can work in the divisible hull $\Phi\otimes_\Z\Q$ of $\Phi$ with the natural extension of the ordering on $\Phi$ and therefore assume that $\Phi$ is the lexicographic product of groups of rank one
 $$\Phi=\Xi_1\times \ldots\times\Xi_h$$
 with $\Psi_j=\{0\}\times\cdots\times\{0\}\times\Xi_{j+1}\times \ldots\times\Xi_h$.\par\noindent
  Let us denote by $r_j$ the rational rank of the group $\Xi_j$.
 

 Now let us choose an ordered embedding of $\Phi\otimes_\Z\Q$ in $(\R^h)_{\hbox{\rm lex}}$. For each $j,\ 1\leq j\leq h$, we can define a vector ${\mathbf w}(j)\in \check\R^N$; it is the vector whose coordinates are the projections in $\Xi_j$ of the $w(u_i)$ viewed as real numbers through the chosen embedding.\par\noindent
 \begin{lemma}\label{rat} Given $N$ elements $w_1,\ldots ,w_N$ of $\R$, the rational rank of the subgroup of  $\R$ generated by the $w_i$ is equal to the dimension of the smallest vector subspace $\langle w\rangle_\Q$ of $\R^N$ defined over $\Q$ containing the vector ${\mathbf w}=(w_1,\ldots ,w_N)$.
 \end{lemma}\label{rankdim}
 \begin{proof} Let $\Ll$ be the kernel of the $\Z$-linear map $b\colon \Z^N\to \R$ sending the $i$-th basis vector to $w_i$. The image of the map $b$ is the subgroup generated by the $w_i$ and by construction the rank of $\Ll$ is the codimension of $\langle w\rangle_\Q$ in $\R^N$. 
 \end{proof}
  
 
 \begin{lemma}\label{linind} The vectors ${\mathbf w}(j),\ 1\leq j\leq h$, of $\check\R^N$ are linearly independant over $\Q$.
 \end{lemma}
 \begin{proof}  Consider the $\Z$-linear map $B\colon (\Z^N)^h\to \R^h$ defined by sending the basis vector $e_i^{(k)}$ to ${\mathbf w}(k)_i$ for $1\leq i\leq N,\ 1\leq k\leq h$. Since each $\Xi_k\subset \R$ is generated by the ${\mathbf w}(k)_i,1\leq i\leq N $ the group $\Phi=\Xi_1\times\cdots\times \Xi_h$ is the image of this map.\par
 The kernel is the lattice ${\mathcal M}$ in $(\Z^N)^h$ determined by the kernels of the maps $\Z^N\to \Xi_k$ sending $e_i^{(k)}$ to ${\mathbf w}(k)_i$ for each $k,\ 1\leq k\leq h$. The rank of this lattice is $N-r_1+\cdots +N-r_h=hN-(r_1+\cdots +r_h)$. \par
 If there was a linear relation with integral coefficients $\sum_{k=1}^hd_k{\mathbf w}(k)=0$, it would give $N$ elements $\sum_{k=1}^hd_ke_i^{(k)},\ 1\leq i\leq N$ of the kernel of $B$, each of which must be rationally independant of ${\mathcal M}$ since it does not imply any relation among the generators of the groups $\Xi_k$. The existence of such a relation contradicts the fact that the rational rank of $\Phi$ is $r_1+\cdots +r_h$.
 \end{proof}
 For each $j,\ 1\leq j\leq h$, let us denote by $S_j$ the smallest vector subspace of $\check \R^N$ defined over $\Q$ and containing the ${\mathbf w}(k),\ 1\leq k\leq j$.\par\noindent
 Notice that all the vector spaces $S_i$ meet the first quadrant $\check \R_{\geq 0}^N$ outside of the origin since ${\mathbf w}(1)$ is in it. By the properties of the lexicographic order, the vector space $S_h$ meets the interior of $\check \R_{\geq 0}^N$, so that we have $\hbox{\rm dim}(S_h\bigcap \R_{> 0}^N)=\hbox{\rm dim}S_h$.\par

 \begin{lemma}\label{ratrank} a) For each $j,\  1\leq j\leq h$ the dimension of the vector space $S_j$ is $\sum_{k=1}^jr_j$.\par\noindent b) Let $\Sigma$ be a regular fan with support $\check\R_{\geq 0}^N$ which is compatible with the vector spaces $S_j$. There exist $N$-dimensional cones $\sigma$ of $\Sigma$ such that ${\mathbf w}(1)\in \sigma$ and for all $j$ the face $\sigma\bigcap S_j$ of $\sigma$ is a cone of maximal dimension in $S_j$.\par\noindent
 c) For such cones $\sigma$, for each $j$ the number of support hyperplanes of $\sigma$ which contain $S_{j-1}\bigcap\sigma$ but do not contain ${\mathbf w}(j)$ is equal to $r_j$.
 \end{lemma}
 \begin{proof} Assertion a) follows directly from Lemmas \ref{rat} and \ref{linind}.\par\noindent
 To prove b) it suffices to remark that since $\Sigma$ is compatible with the $S_j$, for each $j$ the intersection $S_j\bigcap \Sigma$ is a fan of $S_j\bigcap \check\R_{\geq 0}^N$, which is of the same dimension as $S_j$ as we saw above. Since the support of $\Sigma$ is $\check\R_{\geq 0}^N$ this intersection must contain cones of the maximal dimension, which are faces of cones of $\Sigma$. These cones have the required property. \par\noindent Assertion c) is a reformulation of b).
 \end{proof}
 Assume that $\sigma$ is a regular convex cone of dimension $N$ belonging to a regular fan with support $\R^N_{\geq 0}$ which is compatible with the rational vector spaces $S_j,\ 1\leq j\leq h$. Assume that ${\mathbf w}(1)\in \sigma$. Let us denote by $(L_s)_{1\leq s\leq N}$ the hyperplanes  bounding $\sigma$. For each $j$ there is a largest subset $I(j)\subset \{1,\ldots ,N\}$ such that $S_j\subseteq \bigcap_{s\in I(j)}L_s$. By convention we set $I(0)=\{1,\ldots ,N\}$.\par Let us denote by $L_s^{\geq 0}$ the closed half space of $\R^N$ determined by $L_s$ which contains $\sigma$.\par
 \begin{lemma}\label{centering}
 In this situation, the $N$-dimensional regular convex cone $\sigma$ is $w$-centering if and only if the following holds:\par\noindent
 For each $j,\ 0\leq j\leq h-1$, we have ${\mathbf w}(j+1)\in \bigcap_{s\in I(j)}L_s^{\geq 0}$.
 \end{lemma}
 \begin{proof} Since $\sigma$ is regular, the determinant of its generating vectors is $\pm 1$. According to the description $(*)$ of the monomial map associated to $\sigma$, the weights of the $u_i$ uniquely determine the weights of the $y_i$ in $\Phi$ since the determinant is $\neq 0$. Now writing that $w(y_i)$ is $\geq 0$ in the lexicographic product $\Xi_1\times\ldots\times \Xi_h$ reduces exactly to the expression given in the lemma. We observe that the projections in $\Xi_k$ of the valuations of the $(y_i)_{1\leq i\leq N}$ are the barycentric coordinates of the vector ${\mathbf w}(k)$ with respect to the generators of $\sigma$. If all the barycentric coordinates of ${\mathbf w}(1)$ are positive, then all the $w(y_i)$ are also positive and $\sigma$ is $w$-centering. If some of these barycentic coordinates are zero, it means that ${\mathbf w}(1)$ is in a face of $\sigma$ whose linear span is the intersection of the $L_s$ for $s\in I(1)$, by the definition of $I(1)$. Then, in order for the corresponding $w(y_j)$ to be non-negative in $\R^h$, it is necessary that the corresponding barycentric coordinates of ${\mathbf w}(2)$ are $\geq 0$, which is equivalent to the inclusion ${\mathbf w}(2)\in \bigcap_{s\in I(1)}L_s^{\geq 0}$, and so on. The proof of the converse statement is obtained in the same way.
 \end{proof}\noindent
\begin{Remark}: 1) If the group $\Phi$ is of rank one, the condition is simply that the vector ${\mathbf w}(1)$ is in $\sigma$, as we have noted above.\par\noindent
 2) The argument uses only the fact that $\sigma$ is simplicial and $N$-dimensional. In fact, given the monomial map associated to the simplicial cone $\sigma$, for each $j$ the images in the group $\Xi_j$ of the orders $w(y_i)$ are the barycentric coordinates of the vector ${\mathbf w}(j)$ with respect to the vertices of $\sigma$.
\end{Remark}
 %---------------------
 \begin{lemma}\label{quadrant} Keep the notations introduced before Lemma \ref{rankdim}. Let ${\mathbf w}(1),{\mathbf w}(2),\ldots ,{\mathbf w}(h)$ be rationally independant vectors in a $d$-dimensional rational vector subspace $W\subset \R^N$, all lying in $W\bigcap \R_{\geq 0}^N$. Let $\sigma\subset \R_{\geq 0}^N$ be an $N$-dimensional regular cone of a fan $\Sigma$ supported in $\R_{\geq 0}^N$ which is compatible with $W$ and the vector spaces $S_j$ defined above and containing ${\mathbf w}(1)$. Assume that $\sum_{i=1}^h r_i=d$ and that $\sigma\bigcap W$ is of dimension $d$. Then there exists a regular $N$-dimensional cone $\sigma'\subset \R_{\geq 0}^N$ of $\Sigma$ satisfying the conditions of Lemma \ref{centering} and whose intersection with $W$ is of dimension $d$.
 \end{lemma}
 \begin{proof}let $(L_s)_{s\in I}, I=\{1,\ldots ,d\}$ be the collection of the supporting hyperplanes of $\sigma$. By construction there is a largest subset $I(1)\subset I$ such that $S_1= \bigcap_{s\in I(1)}L_s$.  In view of Lemma \ref{ratrank}, c) and the compatibility of $\Sigma$ with $S_1$, it is of cardinality $N-r_1$. By Lemma \ref{linind} we know that ${\mathbf w}(2)$ is not in $\bigcap_{s\in I(1)}L_s$. Let us denote by $I(2)\subset I(1)$ the set $\{s\in I(1)\vert {\mathbf w}(2)\in L_s\}$. For each $s\in I(1)\setminus I(2)$, which is of cardinality $r_2$ by Lemma \ref{ratrank}, c),  we denote by $L_s^{\geq 0}$ the closed half space determined by $L_s$ which contains ${\mathbf w}(2)$. Again by Lemma \ref{linind} we know that ${\mathbf w}(3)\notin \bigcap_{s\in I(2)}L_s$ so we can define a subset $I(3)\subset I(2)$ by the condition that ${\mathbf w}(2)\in \bigcap_{s\in I(3)}L_s$ and a closed half space $L_s^{\geq 0}$ for each $s\in I(2)\setminus I(3)$, and so on. \par In the end we have built a sequence of subsets $$\{1,\ldots ,N\}\supset I(1)\supset I(2)\supset\cdots \supset I(h)$$ such that $S_t=\bigcap_{s\in I(t)}L_s$ and we have determined half spaces $L_s^{\geq 0}$ corresponding to all the hyperplanes $L_s$ for $s\in \{1,\ldots ,N\}\setminus I(h)$ in such a way that each ${\mathbf w}(k)$ is always in the half space $L^{\geq 0}_s$ if $L_s$ vanishes on ${\mathbf w}(k-1)$. According to Lemma \ref{ratrank}, c), at step $i$ we define $r_i$ half-spaces. Since $\sum_{i=1}^hr_i=d$, the set $I(h)$ is the set of those hyperplanes $L_s$ which contain $W$. Thus $\bigcap_{s\in \{1,\ldots ,N\}\setminus I(h)}L_s^{\geq 0}$ is a rational cone which is the intersection of $d$ half spaces in $\R^N$. If now we define for $s\in I(h)$ the half-space $L_s^{\geq 0}$ as the one containing $\sigma$, we see that $\sigma'=\bigcap_{s\in \{1,\ldots ,N\}}L_s^{\geq 0}$ is a rational regular cone which satisfies the conditions of Lemma \ref{centering}; by construction it satisfies the positivity conditions with respect to the ${\mathbf w}(k)$, and its intersection with $W$ is the intersection with $\R^N_{\geq 0}$ of the $d$ half-spaces $(L_s^{\geq 0}\bigcap W)_{s\in \{1,\ldots ,N\}\setminus I(h)}$ of $W$ and is therefore a regular $d$-dimensional cone of the fan $\Sigma\bigcap W$. \noindent
\end{proof}
\begin{remark} If we transform $\sigma$ into the first quadrant by a unimodular transformation, then $\sigma'$ becomes another quadrant.\end{remark}
 
  \begin{proposition}\label{exist}
 Keeping the same notations, let $\Sigma$ be a regular fan with support $\check \R_{\geq 0}^N$. Assume that it is compatible with the $S_j$ and the $H_\ell$. Then there is a cone $\sigma$ of dimension $N$ of $\Sigma$ which is $w$-centering and whose intersection with $W$ is of dimension $d$.  \par\noindent
  \end{proposition}
  %--------------------
 \begin{proof} Let $\sigma$ be a cone of $\Sigma$ containing ${\mathbf w}(1)$. Piltant's theorem tells us that the dimension $d$ of the toric variety defined by the initial binomial ideal, which is also the dimension of the vector space $W$, is equal to the rational rank of the group $\Phi$:
 $$r_1+\cdots +r_h=\hbox{\rm rat.rk} \Phi =d.$$
 It suffices now to apply Lemma \ref{quadrant}.
 
 \end{proof}
\begin{Remark} 1) The key point in the proof of Proposition \ref{exist} is the linear independance over $\Q$ of the ${\mathbf w}(j)$.\par\noindent 
2) We could have proved Proposition \ref{exist} by translating the weights into valuations on the ring $k[u_1,\ldots ,u_N]$ and its quotient by the binomial ideal, and then invoking the valuative criterion of properness. However some work is required to obtain the result in this form, and more importantly this combinatorial proof prepares the extension to the overweight deformations.
\end{Remark}
 Given an overweight deformation as in $(OD)$ above, let us define for each $\ell$ indexing the binomial $u^{m^\ell}-\lambda_\ell u^{n^\ell}$ the following cones in $\R^N$:
 $$\begin{array}{lr}
E_\ell^{(1)}(j)=& \langle \{p-n^\ell/ \vert w (u^{p-n^\ell})\in \Psi_j\setminus\Psi_{j+1}\vert c_p^{(\ell)}\neq 0,\ \}, m^\ell-n^\ell\rangle\\ E_\ell^{(2)}(j)= &\langle \{p-m^\ell/\vert w(u^{p-m^\ell})\in \Psi_j\setminus\Psi_{j+1}\vert c_p^{(\ell)}\neq 0,\ \}, n^\ell-m^\ell\rangle\end{array}$$
  \begin{lemma}\label{strictcon} For all $\ell$ and $0\leq j\leq h-1$ the cone $E_\ell^{(1)}(j)$ (respectively $E_\ell^{(2)}(j)$) is contained in a strictly convex polyhedral rational cone whose elements satisfy  $\langle{\mathbf w}(k),q\rangle= 0$ for $1\leq k\leq j-1$ and $\langle{\mathbf w}(j),q\rangle\geq 0$, with $\langle{\mathbf w}(j),q\rangle=0$ if and only if $q$ is on the half-line generated by $m^\ell-n^\ell$ (respectively $n^\ell-m^\ell$).
\end{lemma}
 %---------------------
\begin{proof} Since the ring $k[[u_1,\ldots , u_N]]$ is noetherian, for each $\ell$ the ideal generated by the monomials $u^p$ appearing in the $\ell$-th series is generated by finitely many of them, say $u^{p^\ell_1},\ldots u^{p^\ell_{s_\ell}}$. In view of the convexity of the subgroups $\Psi_j$ the cones $ E_\ell^{(1)}(j)$ and $E_\ell^{(2)}(j)$  are contained respectively in the convex cones generated by the $p^\ell_k-n^\ell+\R_{\geq 0}^N, \ m^\ell-n^\ell$ and by the $p_k-m^\ell+\R_{\geq 0}^N, \ n^\ell-m^\ell$ (for $1\leq k\leq s_\ell$) for which $w(p-n^\ell)$ (respectively $w(p-m^\ell)$) is in $\Psi_j\setminus \Psi_{j+1}$. These cones are rational since the $p^\ell_k$ are finite in  number and they are strictly convex since they can be defined using strict inequalities and thus cannot contain a vector subspace. The second part of the statement follows from the definition of the vectors ${\mathbf w}(k)$.
\end{proof}
 Since what we want in the end is to find regular convex cones contained in the convex duals $ \check E_\ell^{(i)}(j)$ of the cones $ E_\ell^{(i)}(j)$, we may in view of this lemma assume that the cones $ E_\ell^{(i)}(j)$ themselves are polyhedral rational strictly convex cones, which we shall do henceforth.
\begin{lemma}\label{position}Still denoting by $H_\ell$ the hyperplane of $\check\R^N$ dual to $m^\ell-n^\ell$ and by $W$ the intersection of the $H_\ell$, for each $j$ and each $\ell$ we have :
\begin{itemize}
\item The cones $\check E_\ell^{(1)}(j)$ and $\check E_\ell^{(2)}(j)$ are $N$-dimensional, and their intersection\break $\check E_\ell^{(1)}(j)\bigcap\check E_\ell^{(2)}(j)$ is equal to $\check E_\ell^{(1)}(j)\bigcap H_\ell=\check E_\ell^{(2)}(j)\bigcap H_\ell$.
\item For $i=1,2$ the dimension of $\check E_\ell^{(i)}(j)\bigcap H_\ell$ is $N-1$.
\item The interior in $H_\ell$ of $\check E_\ell^{(1)}(j)\bigcap\check E_\ell^{(2)}(j)$ is contained in the interior of $\check E_\ell^{(1)}(j)\bigcup\check E_\ell^{(2)}(j)$.
\item For each $k$ the cone $\R {\mathbf w}(1)+\cdots +\R {\mathbf w}(k-1)+\R_{\geq 0}{\mathbf w}(k)$ is contained in $\check E_\ell^{(1)}(k)\bigcap\check E_\ell^{(2)}(k)$ and meets its relative interior in $H_\ell$.
\item The same statements are true if one replaces each $\check E_\ell^{(i)}(k)$ by $\bigcap_\ell \check E_\ell^{(i)}(k)$ and $H_\ell$ by $W$.
\end{itemize}

\end{lemma}
 %---------------------
\begin{proof} The dimensionality statement is nothing but the fact that the $ E_\ell^{(i)}(j)$ are strictly convex. An element $a\in \check\R^N$ which is in $\check E_\ell^{(1)}(j)\bigcap\check E_\ell^{(2)}(j)$ has to be both $\geq 0$ and $\leq 0$ on $m^\ell-n^\ell$, so it is in $H_\ell$. Since $p-n^\ell=p-m^\ell +m^\ell-n^\ell$ an element of $H_\ell$ which is $\geq 0$ on $ E_\ell^{(1)}(j)$ is $\geq 0$ on $ E_\ell^{(2)}(j)$ and conversely.\par\noindent
The second statement follows by convex duality from Lemma \ref{strictcon} which implies that $\R \langle m^\ell-n^\ell\rangle$ is the largest vector space contained in $E_\ell^{(i)}(j)+\R \langle m^\ell-n^\ell\rangle$. \par\noindent
The third statement is true because we join two convex cones along a common face of codimension 1; the boundary of the union does not meet the interior of the face.\par\noindent
The fourth statement also follows by convex duality from Lemma \ref{strictcon} if one observes that convex duality, which reverses inclusions, transforms intersection into Minkowski sum. Furthermore, if we add to a vector $a{\mathbf w}(k)$ with $a>0$ a vector $\epsilon {\mathbf b}$ with $b\in H_\ell$, for $\vert \epsilon\vert$ small enough the vector $a{\mathbf w}(k)+\epsilon {\mathbf b}$ will still take positive values on the $p_k-n^\ell$, where the $p^\ell_k$ are defined in the proof of Lemma \ref{strictcon}, and therefore will belong to $\check E_\ell^{(1)}(k)\bigcap\check E_\ell^{(2)}(k)$.\par\noindent
Finally, the same arguments apply to $W$.

\end{proof}

\begin{proposition}\label{approx}
Given an overweight deformation as in $(OD)$, 
there exist regular fans $\Sigma$ with support $\check\R_{\geq 0}^N$ compatible with the hyperplanes $H_\ell$, the vector spaces $S_j$ and all the cones $\check E^{(i)}_\ell(k), i=1,2,\ k=1,\ldots ,h$.\par\noindent
Such fans contain  $w$-centering regular cones $\sigma =\langle a^1,\ldots,a^N\rangle$  contained for all $\ell$ and $k$ in one of the two cones $\check E_\ell^{(i)}(k)\ \ i=1,2$. For such a cone we have $\langle a^i,p-m^\ell\rangle\geq 0$ if $\langle a^i, n^\ell-m^\ell\rangle\geq 0$ (resp. $\langle a^i,p-n^\ell\rangle\geq 0$ if $\langle a^i, m^\ell-n^\ell\rangle\geq 0$) for all monomials $p$ with $w(p)>w(m^\ell)=w(n^\ell)$ appearing in the overweight deformation.
\end{proposition}
 \begin{proof} 
By the resolution theorem for toric varieties we know that there exist regular fans $\Sigma$ with support $\check\R_{\geq 0}^N$ compatible with the $H_\ell$, the $S_j$ and the $\check E^{(i)}_\ell(k)$, which all determine rational cones in $\check\R_{\geq 0}^N$. According to Lemma  \ref{exist} such fans contain $w$-centering cones. Let us show that such a cone is contained in every $\check E^{(1)}_\ell(k)\bigcup \check E^{(2)}_\ell(k)$. In view of the compatibility, it suffices to show that $\sigma$ meets the interior of $\check E^{(1)}_\ell(k)\bigcup \check E^{(2)}_\ell(k)$. To prove that, in view of Lemma \ref{position} it is enough to check that $\sigma$ meets the interior of $\check E^{(1)}_\ell(k)\bigcap H_\ell$ in $H_\ell$.\par
We see that by construction the cone $\sigma$ contains points of the cone $\R {\mathbf w}(1)+\cdots +\R {\mathbf w}(k-1)+\R_{\geq 0}{\mathbf w}(k)$ which are in the interior of $\check E^{(1)}_\ell (k)\bigcap W$ in $W$. Thus the cone $\sigma$ meets the interior of each $\check E^{(1)}_\ell (k)\bigcap \check E^{(1)}_\ell (k)$ and by compatibility of $T$ with the $\check E^{(i)}_\ell (k)$ it is contained in the union $\check E^{(1)}_\ell (k)\bigcup \check E^{(2)}_\ell (k)$. By compatibility with the $H_\ell$ the cone $\sigma$ has to be entirely on one side of $H_\ell$, which means that it must be in 
$\check E^{(1)}_\ell (k)$ or $\check E^{(2)}_\ell (k)$. But this is decided for each $\ell$ by the fact that one generating vector of $\sigma$ is on one side of $H_\ell$.\par
If we write $\sigma=\langle a^1,\ldots ,a^N\rangle$ we see that it has the property that whenever for a given $\ell$ we have that $\langle a^i,m^\ell -n^\ell\rangle$ is $> 0$ for some $i$ then for all $i$ we have $\langle a^i,m^\ell -n^\ell\rangle\geq 0$, and $\langle a^i,p-n^\ell\rangle \geq 0$ for all $p$ appearing in the $\ell$-th equation, and if $\langle a^i,n^\ell -m^\ell\rangle$ is $> 0$ for some $i$, then for all $i$ we have $\langle a^i,n^\ell -m^\ell\rangle\geq 0$ and $\langle a^i,p-m^\ell\rangle\leq 0$ for those $p$. Given $\ell$ there has to be an index $i$ for which $\langle a^i,m^\ell -n^\ell\rangle\neq 0$.
\end{proof}
 %---------------------------------------

 Let us now finish the proof of Proposition \ref{resoverwght}. Take a regular fan $T$ with support $\check\R^N_{\geq 0}$ and compatible with the $H_\ell$, the $S_j$ and the $\check E_\ell^{(i)}\ \ i=1,2$ (and so depending on the deformation), and a $w$-centering cone $\sigma=\langle a^1,\ldots, a^N\rangle$ of that fan as above. Let us write the transforms of the equations $F_1,\ldots, F_s$, with the convention that $y^{\langle a, m\rangle}=y_1^{\langle a^1, m\rangle}\ldots y_N^{\langle a^N, m\rangle}$. 
 $$\begin{array}{lr}
\tilde F_1=y^{\langle a, m^1\rangle}-\lambda_1 y^{\langle a, n^1\rangle}+\Sigma_{w(p)>w(m^1)} c^{(1)}_py^{\langle a, p\rangle}\\
\tilde F_2=y^{\langle a, m^2\rangle}-\lambda_2 y^{\langle a, n^2\rangle}+\Sigma_{w(p)>w(m^2)} c^{(2)}_py^{\langle a, p\rangle}\\
.....\\
\tilde F_\ell=y^{\langle a, m^\ell\rangle}-\lambda_\ell y^{\langle a, n^\ell\rangle}+\Sigma_{w(p)>w(m^\ell)} c^{(\ell)}_py^{\langle a, p\rangle}\\
.....\\
\tilde  F_s=y^{\langle a, m^s\rangle}-\lambda_s y^{\langle a, n^s\rangle}+\Sigma_{w(p)>w(m^s)} c^{(s)}_py^{\langle a, p\rangle}\\
\end{array}
$$
Thanks to the properties of our cone $\sigma$ we may factor out of each $\tilde F_\ell$ either $y^{\langle a, m^\ell\rangle}$ or $y^{\langle a, n^\ell\rangle}$. This leaves us with a deformation of the strict transform of the toric variety, which is regular in the chart corresponding to $\sigma$. More precisely, if for convenience of notation we rearrange the binomials in such a way that all $\langle a^i,m^\ell-n^\ell\rangle$ are $\geq 0$, by writing $n^\ell -\lambda_\ell ^{-1}m^\ell$ if $ \langle a^k, m^\ell -n^\ell\rangle<0$ we can write
$$\begin{array}{lr}
\tilde F_1=y^{\langle a,n^1\rangle} \bigl(y^{\langle a, m^1-n^1\rangle}-\lambda_1 +\Sigma_{w(p)>w(m^1)} c^{(1)}_py^{\langle a, p-n^1\rangle}\bigr)\\
\tilde F_2==y^{\langle a,n^2\rangle} \bigl(y^{\langle a, m^2-n^2\rangle}-\lambda_2 +\Sigma_{w(p)>w(m^2)} c^{(2)}_py^{\langle a, p-n^2\rangle}\bigr)\\
.....\\
\tilde F_\ell=y^{\langle a,n^\ell\rangle} \bigl(y^{\langle a, m^\ell-n^\ell\rangle}-\lambda_\ell +\Sigma_{w(p)>w(m^\ell)} c^{(\ell)}_py^{\langle a, p-n^\ell\rangle}\bigr)\\
.....\\
\tilde  F_s=y^{\langle a,n^s\rangle} \bigl(y^{\langle a, m^s-n^s\rangle}-\lambda_s +\Sigma_{w(p)>w(m^s)} c^{(s)}_py^{\langle a, p-n^s\rangle}\bigr)\\
\end{array}
$$
This end the proof of the Proposition.
 \end{proof} 

 \end{section}
\begin{section}{The complete case}
Now let us go back to the situation and notations of section \ref{localun} and especially of Theorem \ref{cohen}. In this section we assume that $R$ is a complete equicharacteristic local domain. 
 Assume that we are in the situation of Theorem \ref{cohen} and that there exists a finite set $(u_i)_{i\in F}$ of coordinates such that for every $j\in I\setminus F$ there exists among the $F_{mn}$ one, which we denote by $F_j$, in which $u_j$ appears linearly. Let us write for simplicity
$$F_j=u^{n^j}-\lambda_ju^{m^j}+ c_ju_j+\Sigma_pc^{(j)}_pu^p$$
with $c_j\in k^*$, and note that since the weight of $u_j$ is greater that the weight of the  initial binomial, it cannot appear there.\par
By the implicit function theorem (see the Appendix), we know that $u_j$ can be expressed, modulo the closure $I$ in $\widehat{k[(u_i)_{i\in I}]}$ of the ideal generated by the $F_{mn}$, as a series in the $(u_i)_{i\in F}$. We know also that we can invert the equation $F_j$ as
$$u_j=c_j^{(-1)}\bigl(u^{n^j}-\lambda_ju^{m^j}+\Sigma d_pu^p\bigr),$$
an expression where we only know that $u_j$ does not appear on the right-hand side and the terms $d_pu^p$ are of weight greater than the weight of $u^{n^j}$.
\begin{lemma} The images of the $F_{mn}$ other than $F_j$ in the ring $\widehat{k[(u_i)_{i\in I}]}/\bigl(u_j-c_j^{(-1)}\bigl(u^{n^j}-\lambda_ju^{m^j}+\Sigma d_pu^p\bigr)\bigr)\approx \widehat{k[(u_i)_{i\in I\setminus \{j\}}]}$ form a Gr\"obner basis of the ideal which they generate, with respect to the filtration determined by $w(u_i)=\gamma_i$ for $i\neq j$.
\end{lemma}
\begin{proof} We know that the $F_{mn}$ form a Gr\"obner basis of the ideal they generate in $\widehat{k[(u_i)_{i\in I}]}$.  So the initial form of a sum $\Sigma_{mn} A_{mn}F_{mn}$ is in the ideal of $k[(u_i)_{i\in I}]$ generated by the binomials $(u^m-\lambda_{mn}u^n)_{(m,n)\in E}$, where $E$ is the our set of generators of the binomial relations defining $\hbox{\rm gr}_\nu R$. Now we have to see how the initial terms  in $(u_i)_{i\neq j}$ appear when we substitute $u_j$ by $u^{m^j}-\lambda_ju^{n^j}+\Sigma_{w(p)>w(u^{m_j})} d_pu^p$ in a series $\Sigma A_{mn}F_{mn}$. Let us denote by $u'$ the set of variables $(u_i)_{i\neq j}$. In view of the definition of $\widehat{k[(u_i)_{i\in I}]}$ we can write our sum
$$\Sigma_{mn} A_{mn}F_{mn}=\Sigma_{k=0}^\infty D_k(u') u_j^k.$$
Given an element $S\in \widehat{k[(u_i)_{i\in I}]}$, let us  denote by $\tilde S\in  \widehat{k[(u_i)_{i\in I\setminus \{j\}}]}$ the image via the isomorphism $\widehat{k[(u_i)_{i\in I}]}/ (\overline {F_j})\cong \widehat{k[(u_i)_{i\in I\setminus \{j\}}]}$ of the class of $S$ modulo the closure $\overline {(F_j)}$ of the ideal generated by $F_j$. Consider the filtration of $\widehat{k[(u_i)_{i\in I\setminus \{j\}}]}$ determined by giving $u_i$ the weight $\gamma_i$. We denote by $\hbox{\rm In}_w\tilde S\in k[(U_i)_{i\in I\setminus \{j\}}]$ the initial form of $\tilde S$. 
\begin{lemma} a) If the sum $S=\Sigma A_{mn}F_{mn}$ is not contained in $\overline {(F_j)}$, $\hbox{\rm In}_w\tilde S$ is a nonzero element of $k[(U_i)_{i\in I\setminus \{j\}}]$ of the form 
$\Sigma_k M_k(u^{n^j}-\lambda_ju^{m^j})^k$. \par\noindent
b) If we denote by $\hbox{\rm In}_w\tilde I $ the initial ideal in $k[(U_i)_{i\in I\setminus \{j\}}]$ of the ideal generated by the $\tilde{F_{mn}},\ (m,n)\neq (m^j, n^j)$ with respect to the filtration by weight, then there is a nonzero homogeneous $E\in k[(U_i)_{i\in I\setminus \{j\}}]$ such that $E(U^{m^j}-\lambda_j U^{n^j})\in \hbox{\rm In}_w\tilde I $. 
\end{lemma}
\begin{proof}

\par\noindent
To prove b), remark that by definition of $F$, the value group of the valuation is rationally generated by the $(\gamma_j)_{j\in F}$. Since $j\notin F$, among all the relations between the $\gamma_i$, there must be at least one involving $\gamma_j$. Therefore among our binomials there is one of the form $u_j^{m(j)}u'^{m'}-\lambda'u'^{n'}$ where monomials denoted with $u'$ do not contain $u_j$. The corresponding equation is
$$u_j^{m(j)}u'^{m'}-\lambda' u'^{n'}+\Sigma c_{k,q}u'^ku_j^q.$$ It cannot be in the ideal $\overline {(F_j)}$.
Let us estimate its weight in  $\widehat{k[(u_i)_{i\in I\setminus \{j\}}]}$ after the substitution $u_j\mapsto c_j^{(-1)}\bigl(u^{n^j}-\lambda_ju^{m^j}+\Sigma d_pu^p)$. If the lowest weight comes from the initial binomial, it is the weight of $(u^{m^j}-\lambda_ju^{n^j})^{m(j)}u'^{m'}$. The terms $c_{k,0}u'^k$ cannot contribute to the initial form since their weight is greater than the weight in $\widehat{k[(u_i)_{i\in I}]}$ of the initial binomial and is not affected by the substitution, while the weight of the initial binomial is strictly lowered. Therefore all the terms in the initial form are divisible by $U^{m^j}-\lambda_j U^{n^j}$, which proves b).
\end{proof}
\end{proof}
 \end{section}
 \begin{section}{The excellent case}
Then there is the difficulty of reducing the excellent equicharacteristic case to the complete case. This is not entirely settled yet, although there is a precise program to deal with it.
 \end{section}
 \newpage
 \begin{center}
  PART II:  ERRATA AND ADDENDA
\end{center}  
$\bullet$ page 362, table of contents, section 4, read: ...valuation rings and valued n\oe therian rings.\par\noindent
$\bullet$ page 366, line -15, read: ... by Abhyankar's inequality we have $\hbox{\rm r}(\nu)=\hbox{\rm dim}\hbox{\rm gr}_\nu R\leq\hbox{\rm dim}R$.\par\noindent
$\bullet$ page 369, footnote, the last letters in both words of the first Greek
quotation are sigmas. \par\noindent
$\bullet$ page 370 before 2.2 add the following: \textbf{Remark} Given a ring $R$ and a group $\Phi$, the datum of a graded $R$-subalgebra ${\mathcal G}$ of the group algebra $R[v^\Phi]$ is equivalent to the datum of a family of ideals $(I_\phi)_{\phi\in\Phi}$ of $R$ such that $I_\phi.I_\psi\subseteq I_{\phi+\psi}$, the correspondence being described by the equality ${\mathcal G}=\bigoplus_{\phi\in\Phi}I_\phi v^{-\phi}$. If in addition $\Phi$ is a totally ordered group, and $I_\phi=R$ for $\phi\leq 0$, then ${\mathcal G}$ is a graded $R[v^{\Phi_+}]$-algebra if and only if we have $I_\phi \supseteq I_\psi$ whenever $\psi\geq\phi$.
\par\noindent
$\bullet$ page 371, lines 11-13: it is implicitely assumed that each $\overline
x_\phi$ is $\neq 0$.
\par\noindent 
$\bullet$ page 373 line -3, read: ...are respectively isomorphic to $\hbox{\rm Spec}(R\otimes_k(v^{\Phi_+})^{-1}k[v^{\Phi_+}])$ and to $\hbox{\rm Spec}R$.
\par\noindent 
$\bullet$ page 374, line 3, the correct equality is 
$$\Aa_\nu(R)/(\bigoplus_{\phi \in \Phi_+}\Pp_\phi (R)v^{-\phi}+m\Aa_\nu (R))\simeq k[v^{\Phi_+}]$$
\par\noindent 
$\bullet$ page 383, at the end of subsection 3.3, add:\par\noindent
Indeed, except for $\phi=0$, all these ideals are equal to the maximal ideal of $R$, which is also the ideal $m_{\nu_1}\cap R$. The image in $\overline R_1[v^\Phi]=k[v^\Phi]$ of $\A_\Psi (R)$ is equal to $k$. This is as it should be since the valuation $\overline \nu$ is trivial.\par\noindent
$\bullet$ page 384, line -20, read: ``...which can also be deduced from subsection 3.3''
\par\noindent 
$\bullet$ page 389, title of section 4, read: ...valuation rings and valued n\oe therian rings.
\par\noindent 
$\bullet$ page 390, line 14, read: ...with $c_{m,m'}\in k,\ \ \lambda_{m,m'}\in k^*$.\par\noindent 
$\bullet$ page 392, line -14, read: ``...version of (non embedded) local uniformization;''
\par\noindent
$\bullet$ page 393, line 14, read: ( [Z1], B.I, page 861; see also [H-P], Chap. XVIII, \S 5).
\par\noindent
$\bullet$ page 397, line -3, read:..irrational and $>1$,
\par\noindent
$\bullet$ page 398, lines 20,21, read: where $U_1=V_2$ is of degree $1$ and $U_2=V_1$ of degree $\tau$.\par\noindent Addendum:  In the notations of the Perron algorithm, the degree of $V_i$ is $\tau_2^{(i)}$ and the degree of $V_{i+1}$ is $\tau_1^{(i)}$.
\par\noindent
$\bullet$ page 401, line 18, read: $P(u_2,u_3)$. \par\noindent
$\bullet$ page 401, footnote: the third letter of the Greek word is an upsilon
and the last letter  is a sigma.
\par\noindent 
$\bullet$ page 415, lines 4 and 5 of the proof of Lemma 5.25, replace $(X_i)_{i\in I}$ by $(U_i)_{i\in I}$, $cX^a$ by $cU^a$, $X^a$ by $U^a$.
\par\noindent 
$\bullet$ page 415, in Example 5.27, read:
$$x-\widetilde{\hbox{\rm in}}_\nu x=(a-\tilde {a})f^\ell \ \ \hbox{\rm mod.}f^{\ell +1}$$
\par\noindent 
$\bullet$ page 415, last line of the proof of lemma 5.25: replace ``the sequence $x^{(j)}$'' by `` the sequence $x-x^{(j)}$ of elements of $k[(\xi_i)_{i\in I}]$''.\par\noindent
$\bullet$ page 416, line -6, add a $(-1)^n$ before the last sum sign in the formula.
\par\noindent
$\bullet$ page 417
From ``given in advance'', and until ``we may now repeat'',  replace the text by:

Let $s$ be the least integer such that for
infinitely many values of the integer $j$ we have $\nu_s(x-x^{(j+1)})>\nu_s(x-x^{(j)})$. If $s=1$,
since the valuation $\nu_1$ is of height one, for any $\phi_1\in \Phi_{1+}$, after finitely
many steps we have $\nu_1 (x-x^{(j)})>\phi_1$, hence for any $\phi\in \Phi_{+}$, after finitely
many steps we have $\nu(x-x^{(j)})>\phi$, so we may take $r_j=x-x^{(j)}$ and get $\nu
(x-\sum_{r=0}^{j}P_r(\xi_i))= \nu (x-r_j)>\phi$ and this shows that $x\in \hat R_h$. Assume now $s>1$, and let $x^{(1)}$ be the representative in $\hat R_{s-1}[(\xi_i)_{i\in I_s}]$ of the initial form $\hbox{\rm in}_{\nu_{s-1}}(x)\in \hbox{\rm gr}_{\nu_{s-1}} R $. This makes sense because we may assume by induction on the dimension of $R$ that the subring $\hat R_{s-1}\subset R$ is a representative of  $R/(m_{\nu_{s-1}}\cap R)$. We define inductively the $x^{(j)}$ as representatives in $\hat R_{s-1}[(\xi_i)_{i\in I_s}]$ of $\hbox{\rm in}_{\nu_{s-1}}(x-x^{(j-1)})$. By
definition of $s$, there exists a $j_0$ such that $\nu_{s-1}(x-x^{(j+1)})= \nu_{s-1}(x-x^{(j)})$ for $j\geq
j_0$. By substracting from $x$ a polynomial in the $\xi_i$, we may assume that
$j_0=1$.\par\noindent Note that the center of $\nu_{s-1}$ is necessarily distinct from the
center of $\nu_s$, since if the two centers are equal, there are only finitely many distinct
$\nu_s$-ideals between two consecutive $\nu_{s-1}$-ideals of $R$, according to [Z-S], Vol.
II, Appendix 3, Corollary p.345. \par\noindent We consider the initial forms
$\hbox{\rm in}_{\nu_{s-1}}(x- x^{(j)})\in \hbox{\rm gr}_{\nu_{s-1}} R $; for $j\geq j_0$; they all
have the same degree, say $\phi_{s-1}$ and they are in $\Pp_{\phi_{s(j)}}/\Pp^+_{\phi_{s-1}}$ with $\phi_s(j)$ increasing strictly with $j$. The $R/{\bf p}_{s-1}$-submodules $\Pp_{\phi_s}/\Pp^+_{\phi_{s-1}}\subset \Pp_{\phi_{s-1}}/\Pp^+_{\phi_{s-1}}$ form a simple infinite sequence in view of Proposition 3.17. This sequence of submodules has intersection $(0)$ and by the module-theoretic version of Chevalley's theorem ([B3], Chap. IV, \S 2, No.5, Cor.4) there is a sequence of integers $t(\phi_s(j))$ tending to infinity with the image of $\phi_s(j)$ in the height one group $\Psi_{s-1}/\Psi_s$ and such that $\Pp_{\phi_s(j)}/\Pp^+_{\phi_{s-1}}\subset m^{t(\phi_s(j))}(\Pp_{\phi_{s-1}}/\Pp^+_{\phi_{s-1}})$. Since each
homogeneous component $\Pp_{\phi_{s-1}}/\Pp^+_{\phi_{s-1}}$ of $\hbox{\rm gr}_{\nu_{ s-1}} R $ is a $R/(m_{\nu_{s-1}}\cap
R)$-module of finite type, it is complete for the $m/(m_{\nu_{s-1}}\cap R)$-topology.
It is also complete for the $\overline\nu_s$ topology by (the proof of) Proposition 5.10. The sequence of the initial forms $\hbox{\rm in}_{\nu_{s-1}}(x-x^{(j)})$ then converges in $(\hbox{\rm gr}_{\nu_{s-1}}
R)_{\phi_{s-1}}$ for the $m$-adic topology, and therefore also for the topology defined by the $\Pp_{\phi_s}/\Pp^+_{\phi_{s-1}}$,  to a unique limit $\overline {x_1^{(1)}}$. By the definition of $\hat R_s$, we can lift  $\overline {x_1^{(1)}}$ to an element
$x_1^{(1)}\in
\hat R_s$. This element has the property that for all $j\geq 1$, we have $\nu_{s-1}(x-x_1^{(1)})>\nu_{s-1}(x-x^{(j)})=\nu_{s-1}(x)$. Replacing now $x$ by $x-x_1^{(1)}$, and repeating this construction, we build a sequence of elements
$x_1^{(q)}\in \hat R_h$ such that $\nu_{s-1} (x-x_1^{(q)})$ increases at each step. 
\par\noindent 
$\bullet$ page 418, from the beginning of 5.3 to: ``This is a $\nu$-adic..'', replace the text by:\par\noindent
Given a system $(\overline \xi_i)_{i\in I}$ of generators of the $k$-algebra
$\hbox{\rm gr}_\nu R$, giving rise to a surjective map $k[(U_i)_{i\in I}]\to \hbox{\rm gr}_\nu R$, a field of representatives $k\subset \hat
R^{(\nu)}$ and representatives
$(\xi_i)_{i\in I}$ in $R$ of the $\overline\xi_i$, the main result of this subsection is the extension of the map $k[(u_i)_{i\in I}]\to \hat R^{(\nu)}$ mapping $u_i$ to $\xi_i$ to a continuous
surjective map
$$\widehat{k[(w_j)_{j\in J}]}\to \hat R^{(\nu)}$$ from the scalewise completion of the polynomial ring to the scalewise $\nu$-adic
completion of $R$. When $\widehat{k[(w_j)_{j\in J}]}$ is endowed with the term order obtained by giving to $w_j$ the valuation of its image
$\eta_j\in \hat R^{(\nu)}$, the associated graded map is the map 
$$k[(W_j)_{j\in J}]\to \hbox{\rm gr}_{\hat\nu} \hat R^{(\nu)}$$
mapping $W_j$ to $ \overline\eta_j$, where each $ \overline\eta_j$ is a Laurent monomial in the $\overline\xi_i$. 
\par\noindent
$\bullet$ page 424, replace line 5 by: Using the fact that $\widehat{\Aa_\nu (R)}^{(\nu_\A)}$ is a faithfully flat $k[v^{\Phi_+}]$-algebra, one can show:
\par\noindent 
$\bullet$ page 425, replace line 6 of `Corollary 5.51 by:
$$\hat c\colon \widehat{k[(w_j)_{j\in J}]}
\to \hat R^{(\nu )} ,\ \ \ w_j\mapsto c(-\hat\nu(\eta_j))\eta_j$$\par\noindent
$\bullet$ page 426, first line of the proof of Cor. 5.53: delete ''of
the''.\par\noindent
$\bullet$ page 426, change the statement of Cor. 5.54 to:\par\noindent
{\it Let $R$ be a complete noetherian local ring endowed with a rational valuation $\nu$ of height $\hbox{\rm h}(\nu)$. In the minimal system of
generators of the semigroup $\Gamma$ of $(R,\nu)$ there are at most $\hbox{\rm h}_R(\nu)-1$ elements without predecessor.}
\par\noindent
$\bullet$ page 426, in the proof of Corollary 5.54, from ''All the monomials'' to the end, replace the text by:\par\noindent
Finitely many of these monomials ${\xi''}^{\alpha''}$ generate the
$R/{\mathbf p}$-module $\Pp_{\phi_1}/\Pp^+_{\phi_1}$; denote them by ${\xi''}^{\alpha''_\ell}$. If a linear combination $\Sigma_\ell a_\ell {\xi''}^{\alpha''_\ell}$ of ~$\nu_1$ valuation $\phi_1$ has $\nu$ valuation greater than the minimum of the $\nu$-valuations of its terms, it must contain binomials ${\xi'}^{\alpha'}{\xi''}^{\alpha''_i}-\lambda_{ij}{\xi'}^{\alpha'_j}{\xi''}^{\alpha''_j}$. But such a binomial can be written as  a series beginning with a monomial ${\xi'}^{\alpha'_k}{\xi''}^{\alpha''_k}$. After making all such substitutions we obtain an expression for our sum in which the $\nu$ valuation of each term has increased. After finitely many steps we reach the situation where the $\nu$-valuation of the sum is the minimum of the valuations of its terms,  and this shows that the $\nu$-valuation of every element of $\Pp_{\phi_1}\setminus
\Pp^+_{\phi_1}$ is a linear combination with coefficients in $\N\cup\{0\}$ of elements of the semigroup $\overline \Gamma$ of the residual valuation
$\overline\nu$ and the valuations of the monomials ${\xi''}^{\alpha''_\ell}$.
\par\noindent
$\bullet$ page 426, first line of the proof of Cor. 5.54, read:..height $\hbox{\rm h}_R(\nu)$ of $\nu$ in $R$.
\par\noindent
$\bullet$ page 426, at the end of the proof of Cor. 5.54, read ..height of $\nu$ in $R$ minus one.
\par\noindent
$\bullet$ page 426, line -2, replace ``fact'' by ``condition'' 
\par\noindent
$\bullet$ page 427 line -4, replace $\partial_{u_{h+1}}$ by $\partial_{w_h}$.
\par\noindent
$\bullet$ page 430
\par\noindent
$\bullet$ page 436, lines -12, -13: remove the sentence ``Conversely,...$\{1,\ldots, L\}$''.
\par\noindent
$\bullet$ page 440 line 1: ...of the canonical basis vectors...
\par\noindent
$\bullet$ page 440 in the firs paragraph, after ``..first quadrant of ${\mathbf R}^N$.'', add:\par\noindent
Conversely if $\tilde f (e_i)\geq 0$ and $\tilde f \in W$ then $\tilde f $ is in the image of $\sigma$. This shows that $\sigma$ is exactly the intersection of $W$ with the first quadrant of ${\mathbf R}^N$.
\par\noindent
$\bullet$ page 444 line -15, read : $\hbox{\rm Spec}k[v][[u_1,\ldots ,u_N]]$\par\noindent
$\bullet$ page 449 add at the bottom of the page: One difference between example 5.7 and example 4.20 is that the first one is weakly Abhyankar while the second one is not since the completion of $R$ in this second case is $k[[u_1,u_2]]$.
\par\noindent
$\bullet$ page 451, first line of the statement of Corollary 7.7, read : $\hbox{\rm gr}_{\hat\nu} \hat R^{(\nu)}$ and not $\hbox{\rm gr}_\nu\hat R^{(\nu)}$.
\par\noindent 
$\bullet$ page 457, after [H2], add \par\noindent 
[H-P]  W.V.D. Hodge and D. Pedoe, {\it Methods of Algebraic Geometry, Vol. 3}, Cambridge mathematical library, Cambridge University Press.
\par\noindent 
$\bullet$ page 457,
reference [Kr], the year is 1932.
 
\begin{center}
 (CNRS, Institut math\'ematique de Jussieu, UMR 7586 du CNRS )
  \end{center} 
  \end{document}
  
       DEBRIS
       
  Let $(u^{m^j}-\lambda_j u^{n^j})_{j\in J} $ be a finite or countable collection of binomials in a ring $S$ which is either $k[(u_i)_{i\in I}]$ or its scalewise completion $\widehat{k[(u_i)_{i\in I}]}$ with respect to a weight function $w(u_i)=\gamma_i$ whose values form a minimal system of generators of a well ordered semigroup $\Gamma$ of $\Phi_+$. Fix for each index $j\in J$ an element $P_j\in S$ which is of weight $w(P_j)>w(u^{m^j})=w(u^{n^j})$. Take variables $v^{\gamma_j}$ indexed by $J$ and set  $T=k[(v^{\gamma_j})_{j\in J}]$ and let $K$ be the closure of the ideal of $\A= T\otimes_k S$ generated by the elements $(u^{m^j}-\lambda_j u^{n^j}-v^{\gamma_j}P_j)_{j\in J}$. Denote by $R$ the algebra $\A/(v^\phi-1)_{\phi\in \Phi_+}=S/K_1$, where $K_1$ is the ideal obtained from $K$ by setting all $v_j=1$ since the $v^{\gamma_j}$ generate the semigroup of the $v^\phi;\ \phi\in \Gamma$. 
\begin{proposition}The following conditions are equivalent:\par\noindent a)The weight $w$ induces a valuation $\nu$ on the ring $R$, such that the image of $u_i$ has value $\gamma_i$.\par\noindent b) The $T$-algebra $(T\otimes_k S)/K$ is faithfully flat. \par If that is the case $(T\otimes_k S)/K$ coincides with the valuation algebra of $(R,\nu)$.\end{proposition}
\begin{proof} For $x\in S_1$ define $\nu(x)$ to be the maximum weight of an element of $S$ whose image in $R$ is $x$. Using the fact that weights give a monomial order on $S$ we see that $\nu (x+y)\geq\hbox{\rm min}(\nu(x),\nu(y))$ and 
Set $N=\vert F\vert$, and let $\Sigma$ be a regular fan refining the first quadrant of $\check \R^N$ and \textit{compatible} with the hyperplane $H_{m-n}$ dual to the vector $m-n\in \R^N$. Such fans always exist (see [E]). The compatibility condition means that if $\sigma=\langle a^1,\ldots , a^N \rangle$ is a regular rational cones of $\Sigma$ which is of dimension $N$, generated by the vectors $a^j$, then the intersection $\sigma\cap H_{m-n}$ is a face of the convex cone $\sigma$ (which may be $\{0\}$).\end{proof}
\begin{lemma}
 Given a weight function $w$ on a polynomial ring $k[u_1,\ldots, u_N]$ or a power series ring, with positive values in a totally ordered group $\Phi$ of finite rank, there exists a vector ${\mathbf w}\in \R_{\geq 0}^N$ such that $w(u^m)\geq w(u^n)$ if and only if $\langle{\mathbf w},m\rangle \geq \langle{\mathbf w},n\rangle$. This vector is unique up to multiplication of each coordinate by a positive real number.\end{lemma}
We can embed the group $\Phi$ in its divisible hull $\Q^r$, where $r$ is the rational rank of $\Phi$, and extend the ordering by the condition that $\eta\leq \zeta$ in $\Q^r$ if there exists a positive integer $d$ such that $d\eta \leq d\zeta$ in $\Phi$. Then (see [Ab]), the totally ordered group $\Q^r$ decomposes as a direct sum of groups of height one: $\Q^r=\Xi_1\bigoplus\cdots \bigoplus \Xi_h$. The ordering on $\Q^r$ is then lexicographic with the $\Xi_k\bigoplus\cdots \bigoplus \Xi_h$ with $ 2\leq k\leq h$ as non trivial convex subgroups.\par In this decomposition, for each index $i$ let $k(i)$ be the smallest index $k$ such that the image of $w(u_i)$ in $\Xi_k$ by the natural projection is not zero. Let us denote by ${\mathbf w}(u_i)$ this image.\par\noindent
 The embedding in $\R$ of each $\Xi_k$ is unique up to multiplication by a positive factor and since we are interested only in the consequences of inequalities such as $w(p-n)>0$, this non-uniqueness is of no import. Let us therefore pick one ordered embedding of each $\Xi_k$ into $\R$ and consider the vector ${\mathbf w}\in \R^N$ with coordinates ${\mathbf w}_i={\mathbf w}(u_i)$.
\par\noindent For $1\leq \ell\leq s$, let $C_\ell^{(1)}$ be the convex cone with vertex $0$ generated by the differences $p-n^\ell$ as $p$ runs through the exponents appearing in the deformation of the $\ell$-th binomial, including $m^\ell$. Let $C_\ell^{(2)}$ be the cone with vertex $0$ generated by the differences $p-m^\ell$ as $p$ runs through the exponents appearing in the deformation of the $\ell$-th binomial, including $n^\ell$.
\begin{lemma}\label{strictcon} For all $\ell$ the cones $C_\ell^{(1)}$ and $C_\ell^{(2)}$ are contained in strictly convex polyhedral rational cones $ F_\ell^{(1)}$ and $ F_\ell^{(2)}$ whose elements satisfy $w(q)\geq 0$, and $w(q)=0$ if and only if $q$ is on the line generated by $m^\ell-n^\ell$.
\end{lemma}
\begin{proof} Let us first show that given any order function on $k[u_1,\ldots ,u_N]$ (or $k[[u_1,\ldots ,u_N]]$), we can associate to it a weight vector ${\mathbf w}\in \R^N$ which plays for positivity questions the same role as the usual weight vector, although it does not in general determine the weight function if the rank of the value group is $>1$.  It lies in $W\bigcap \R^N_+$ by construction and it has the property that for all $\ell$ and $p\neq m^\ell,n^\ell$ we have the inequality $\langle {\mathbf w},p-n^\ell\rangle>0$. \par
The semigroup of values of the weight $w$ is well ordered since the ring $S$ is noetherian. For each $\ell$ there is therefore a positive minimal value of $\langle {\mathbf w},p-n^\ell\rangle$ as $p$ runs through the monomials appearing in $F_\ell$ and distinct from $m^\ell$ and $n^\ell$. If we denote by $C_\ell^{(1)}$ (resp. $C_\ell^{(2)}$) the cone generated by these $p-n^\ell$ (resp. $p-m^\ell$), we see that it is strictly convex since it is strictly contained in the half-space $\langle {\mathbf w},q\rangle >0$ and in fact the convex dual $\check C_\ell^{(i)}$ of $C_\ell^{(i)}$ contains the vector ${\mathbf w}$ in its interior for $i=1,2$.
The dual cones $\check C_\ell^{(i)}\ \ i=1,2$ are convex rational cones with non-empty interior in $\check \R^N$. Let us remark that because of their definition, we have  $\check C_\ell^{(1)}\bigcap H_\ell =\check C_\ell^{(2)}\bigcap H_\ell=\check C_\ell^{(1)}\bigcap\check C_\ell^{(2)}$ for all $\ell$.\par
 We can choose polyhedral rational convex cone $\check E_\ell^{(i)},\ i=1,2$ contained in $\check C_\ell^{(i)}$, containing ${\mathbf w}$ in their interiors and satisfying the equalities $\check E_\ell^{(1)}\bigcap H_\ell =\check E_\ell^{(2)}\bigcap H_\ell=\check E_\ell^{(1)}\bigcap\check E_\ell^{(2)}$ for all $\ell$.. The polyhedral rational convex cone $ F_\ell^{(1)}$ (resp. $ F_\ell^{(2)}$) generated in $\R^N$ by the dual $ E_\ell^{(1)}$ of $\check E_\ell^{(1)}$  and $m^\ell -n^\ell$ (resp. by $ E_\ell^{(2)}$ and $n^\ell -m^\ell$) is still strictly convex since the only vector spaces it might contain are in the hyperplane $\langle {\mathbf w}, q\rangle=0$ and its intersection with it is a half-line.
This proves the result.
\end{proof}

