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\begin{document}
\title{Monomial ideals, binomial ideals,\hskip2truecm polynomial ideals}
\author{Bernard Teissier}
\maketitle
\centerline {Lectures at MSRI, September 2002}
%\date{}
\pagestyle{myheadings}
\markboth{\rm Bernard Teissier}{\rm Monomial ideals, binomial ideals,...}
%\maketitle
\par\vskip.4truecm

\centerline{\textbf{Summary}}
These lectures are meant to provide a glimpse of the applications of toric geometry to singularity theory. They will illustrate some ideas and results of commutative algebra by showing the form which they take for very simple ideals of polynomial rings:  monomial or binomial ideals, which can be understood combinatorially. Some combinatorial facts are the expression for monomial or binomial ideals of general results of commutative algebra or algebraic geometry such as resolution of singularities or the Brian\c con-Skoda theorem. In the opposite direction, I will show that there are methods
which allow one to prove results about fairly general ideals by continuously specializing them to monomial or binomial ideals. 
\tableofcontents
\section{Introduction}
\par\medskip
Let $k$ be a field, and $k[u_1,\ldots ,u_d]$ (resp $k[[u_1,\ldots ,u_d]]$) the polynomial (resp. power series) ring in $d$ variables. \par If $d=1$,
given two monomials $u^m, u^n$ , one divides the other, so that if $m>n$ say, a binomial $u^m-\lambda u^n=u^n(u^{m-n}-\lambda)$ with $\lambda\in k^*$ is,
viewed now in
$k[[u]]$, a monomial times a unit. For the same reason any series $\sum_if_iu^i\in k[[u]]$ is the product of a monomial $u^n,\ n\geq 0$, by a unit of $k[[u]]$.
Staying in $k[u]$, we can also view our binomial as the product of a monomial and a cyclic polynomial $u^{m-n}-\lambda$.
\par For $d=2$, working in $k[[u_1,u_2]]$ we meet a serious difficulty:  a series in two variables does not necessarily have a dominant term,
that is, a term which divides all the others. The simplest example is the binomial $u_1^a-cu_2^b$ with $c\in k^*$. As we shall see, if we allow enough
transformations, it is essentially the only example in dimension $2$. So the behavior of a series $f(u_1,u_2)$ near the origin does not reduce to that of the product of a monomial $u_1^au_2^b$ by a unit.
\par\noindent
In general, for $d>1$ and for a given $f(u_1,\ldots ,u_d)\in
k[[u_1,\ldots ,u_d]]$, say
$f=\sum_m f_mu^m$, where
$m\in {\mathbf Z}_{\geq 0}^d$ and $u^m=u_1^{m_1}\ldots u_d^{m_d}$, we can try to measure how different it is from a monomial times a unit by considering,
the ideal of $k[[u_1,\ldots ,u_d]]$ or $k[u_1,\ldots ,u_d]$ generated by the monomials
$\{u^m/f_m\neq 0 \}$ which actually appear in $f$. Since both rings are n\oe therian, this ideal is finitely generated in both cases, and we are faced
with the following problem:\par\noindent
\textbf{Problem}: Given an ideal generated by finitely many monomials (a \textit {monomial ideal}) in $k[[u_1,\ldots ,u_d]]$ or $k[u_1,\ldots ,u_d]$,
study how far it is from being principal.  \par\noindent
We shall also meet a property stronger than principality: given a finitely generated ideal in a ring, the property that given any pair of its generators, one
divides the other implies principality (exercise), but is stronger in general (take an ideal in a Principal Ideal Domain, such as ${\mathbf Z}$ or a
(non-monomial) ideal in
$k[u]$); I shall call it
\textit{strong principality}. The integral domains in which every finitely generated ideal is strongly principal are known as \textit {valuation
rings}. Most of them are not n\oe therian.
\par\noindent Here we reach a bifurcation point in methodology: \par\noindent 
-one approach is to generalize the notion of divisibility by studying all linear relations,
with coefficients in the ambient ring, between our monomials. This leads to the construction of syzygies for the generators of our monomial ideal
${\mathcal M}$, or free resolutions for the quotient of the ambient ring by ${\mathcal M}$. There are many beautiful results in this direction. See
Eisenbud's lectures and [St]. One is also led to try and compare monomials using \textit{monomial orders} to produce Gr\"obner bases, since as soon as the
ideal is not principal, deciding whether a given element belongs to it becomes arduous in general. \par\noindent
- Another approach is to try and force the ideal ${\mathcal M}$ to become principal after a change of variables. It is the subject of the next section.
\par\noindent
\section{Strong principalization of monomial ideals by toric maps}\label{strongp}
 In order to make a monomial ideal principal by changes of variables, the first thing to try is changes of
variables which transform monomials into monomials, that is, which are themselves described by monomial functions:
$$
\begin{array}{llrcl}
u_1 &=& {y_1}^{a_{1}^1} \cdot\cdot\cdot\cdot {y_d}^{a_{1}^d} \\
 u_2 &=& {y_1}^{a_{2}^1} \cdot\cdot\cdot\cdot {y_d}^{a_{2}^d} \\
\cdot &&\cdot && \\
\cdot &&\cdot && \\
\cdot &&\cdot && \\
 u_d &=& {y_1}^{a_{d}^1} \cdot\cdot\cdot\cdot {y_d}^{a_{d}^d}
\end{array}
$$
where we can consider the exponents of $y_i$ appearing in the expressions of $u_1,\ldots ,u_d$ as the coordinates of a vector $a^i$ with integral
coordinates. These expressions decribe a monomial, or \textit{toric}, map of $d$-dimensional affine spaces
$$\pi(a^1,\ldots ,a^d)\colon {\mathbf A}^d(k)\to {\mathbf A}^d(k)$$ in the coordinates $(y_i)$ for the first affine space and $(u_i)$ for the second.\par\noindent
If we compute the effect of the change of variables on a monomial $u^m$, we see that $$u^m\mapsto y_1^{\langle a^1,m\rangle}\ldots y_d^{\langle
a^d,m\rangle}$$
\textbf{Exercise}: Show that the degree of the fraction field extension $k(u_1,\ldots ,u_d)\to k(y_1,\ldots ,y_d)$ determined by $\pi(a^1,\ldots ,a^d)$ is
equal to the absolute value of the determinant of the vectors $(a^1,\ldots ,a^d)$. In particular it is equal to one (that is, our map $\pi(a^1,\ldots ,a^d)$ is
birational) if and only if the determinant of the vectors $(a^1,\ldots ,a^d)$ is
$\pm 1$ that is, $(a^1,\ldots ,a^d)$ are a basis of the integral lattice ${\mathbf Z}^d$.
\par\noindent
\par\medskip\noindent
In view of the form of the transformation on monomials by our change of variables, it makes sense to introduce a copy of ${\mathbf Z}^d$ where the exponents of
our monomials dwell, and which we will denote by $M$, and a copy of ${\mathbf Z}^d$ in which our vectors $a^j$ dwell which we will call the \textit{weight
space} and denote by $W$. The lattices $M$ and $W$ are dual and we consider $W$ as the integral lattice of the vector space $\check{\mathbf R}^d$ dual to the
vector space ${\mathbf R}^d$ in which our monomial exponents live. In this manner, we think of $m\mapsto\langle a^i,m\rangle$ as the linear form on $M$
corresponding to $a^i\in W$.\par Given two monomials $u^m$ and $u^n$, the necessary and sufficient condition for the transform of $u^n$ to divide the
transform of $u^m$ in $k[y_1,\ldots ,y_d]$ is that $\langle a^i,m\rangle \geq\langle a^i,n\rangle\ \ \hbox{\rm for all}\ i,1\leq i\leq d $. If we read this
as
$\langle a^i,m-n\rangle\geq 0\ \hbox{\rm for all}\ i,1\leq i\leq d $ and seek a symmetric formulation, we are led to introduce the rational hyperplane
$H_{m-n}$ in $\check{\mathbf R}^d$ which is dual to the vector $m-n\in M$ and obtain the following elementary but fundamental fact, where the transform
of a monomial is nothing but its composition with the map $\pi (a^1,\ldots ,a^d)$ in the coordinates
$(y_1,\ldots ,y_d)$:
\begin{lemma} \label{div}
 A necessary and sufficient condition for the transform of one of the monomials $u^m,u^n$ by the map $\pi (a^1,\ldots ,a^d)$ to divide the transform
of the other in
$k[y_1,\ldots ,y_d]$ is that all the vectors $a^j$ lie on the same side of the hyperplane $H_{m-n}$ in $\check{\mathbf R}_{\geq 0}^d$.
\end{lemma}
\par\noindent Remark that the condition is non empty if and only if one of the monomials $u^m,u^n$ does not already divide the other
in
$k[u_1,\ldots ,u_d]$, because to say that such divisibility does not occur is to say that the equation of the hyperplane $H_{m-n}$ does not have all its
coefficients of the same sign, and therefore separates in two regions the first quadrant $\check{\mathbf R}_{\geq 0}^d$ where our vectors $a^j$ live.\Par
To force one monomial to divide the other in the affine space ${\mathbf A}^d(k)$ with coordinates $(y_i)$ is nice but not extremely useful since it provides
information on the original monomials only in the image of the map $\pi (a^1,\ldots ,a^d)$ in the affine space ${\mathbf A}^d(k)$ with coordinates 
$(u_i)$, which is
a constructible subset different from ${\mathbf A}^d(k)$. It is much more useful to find a proper and birational (hence surjective) map of algebraic
varieties over $k$
$\pi\colon Z\to
{\mathbf A}^d(k)$ such that the compositions with
$\pi$ of our monomials generate a sheaf of ideals in $Z$ which is \textit{locally} principal; if you prefer, $Z$ should be covered by affine charts $U$ such that
if our monomial ideal ${\mathcal M}$ is generated by $u^{m^1},\ldots , u^{m^q}$, the ideal $(u^{m^1}\circ\pi,\ldots , u^{m^q}\circ\pi)\vert U$ is principal,
or strongly principal.\par\medskip
Toric geometry provides a way to do this: \par\noindent A rational \textit{fan} with 
 $\check{\mathbf R}_{\geq 0}^d$ as support is a finite collection $\Sigma$ of strictly convex rational
cones
$(\sigma_\alpha)_{\alpha\in A}$ with the following properties:\Par
1) The union of all the $(\sigma_\alpha)_{\alpha\in A}$ is the closed first quadrant $\check{\mathbf R}_{\geq 0}^d$ of $\check{\mathbf R}^d$.\Par
2) Each face of a $\sigma_\alpha\in \Sigma$ is in $\Sigma$; 
%ERRATUM un \sigma transforme en \Sigma ligne ci-dessus et in particular  supprime
\Par
3) each intersection $\sigma_\alpha\cap\sigma_\beta$ is a face of $\sigma_\alpha$ and of $\sigma_\beta$.\par\noindent
 In general, the \textit{support} of a fan $\Sigma$ is defined as $\bigcup_{\alpha\in A}\sigma_\alpha$.\par
Recall that a cone is \textit{strictly convex} if it contains no positive dimensional vector subspace, and cones contained in the first quadrant must
be strictly convex.\par\noindent A cone $\sigma\subset \check{\mathbf R}^d$ is strictly convex if and only if its convex dual
$$\check\sigma=\{m\in{\mathbf R}^d/\langle m,a\rangle\geq 0 \ \ \hbox{\rm for all}\ a\in \sigma\}$$ has a non empty interior in ${\mathbf
R}^d$.\par A rational convex cone is one bounded by finitely many hyperplanes whose equations have rational (or equivalently, integral) coefficients. An
equivalent definition is that a rational convex cone is the cone positively generated by finitely many vectors with integral coordinates.\par A fan is said to be
\textit{regular} if each of its $k$-dimensional cones is generated by $k$ integral vectors (it is a simplicial cone) which form part of a basis of the integral lattice.
If $k=d$ it means that their determinant is $\pm 1$.
\par If we go back to our monomial map, assuming that the determinant of the vectors $(a^1,\ldots ,a^d)$ is $\pm 1$, we can express the $y_j$ as
monomials in the $u_i$; the matrix of exponents will then be the inverse of the matrix $(a^1,\ldots ,a^d)$, and have some negative entries. Monomials
with possibly negative exponents will be called \textit{Laurent monomials} here.\Par If $\sigma=\langle a^1,\ldots ,a^d\rangle$, the cone positively
generated by the vectors $a^1\ldots ,a^d$, then the monomials in
$y_1,\ldots ,y_d$, viewed as Laurent monomials in $u_1,\ldots u_d$ via the expression of the $y_j$ as Laurent monomials in the $u_i$, correspond to the
integral points of the dual convex cone $$\check\sigma=\{x\in {\mathbf R}^d/\ell(x)\geq 0\ \forall \ell\in\sigma\}$$ of $\sigma$, i.e., those points $m\in {\mathbf Z}^d$
such that
$\langle a^i,m\rangle \geq0\ \hbox{\rm for all}\ 1\leq i\leq d$. So we can identify the polynomial algebra $k[y_1,\ldots ,y_d]$ with the algebra $k[\check\sigma\cap M] $ of the
semigroup
$\check\sigma\cap M$ with coefficients in $k$. Since $\sigma$ is contained in the first quadrant of $\check{\mathbf R}^d$, its convex dual $\check\sigma$
contains the first quadrant of ${\mathbf R}^d$, so we have a graded inclusion of algebras
$$k[{\mathbf R}_{\geq 0}^d\cap M]=k[u_1,\ldots ,u_d]\subset k[\check\sigma\cap M]=k[y_1,\ldots ,y_d],$$ the inclusion being described by sending each variable
$u_i$ to a monomial in $y_1,\ldots ,y_d$ as we did in the beginning.\par\noindent This slightly more abstract formulation has the following use: Given a fan
in
$\check{\mathbf R}^d$, to each of its cones $\sigma$ we can associate the algebra
$ k[\check\sigma\cap M]$, even if the strictly convex cone $\sigma$ is not generated by $d$ vectors with determinant $\pm 1$.\Par By a lemma of Gordan
(see [TE]), it is finitely generated, so it corresponds to an affine algebraic variety
$X_\sigma=\hbox{\rm Spec}\ k[\check\sigma\cap M]$. This variety is a $d$-dimensional affine space if and only if the cone $\check\sigma$ (or $\sigma$) is
$d$-dimensional and generated by vectors which form a basis of the integral lattice of $\check{\mathbf R}^d$. It is, however, always
normal and with rational singularities only (see [TE]) and moreover it is rational, which means that the field of fractions of $k[\check\sigma\cap M]$ is
$k(u_1,\ldots ,u_d)$.\par  If two cones
$\sigma_\alpha$ and
$\sigma_\beta$ have a common face
$\tau_{\alpha\beta}$, the affine varieties $X_{\sigma_\alpha}$ and $X_{\sigma_\beta}$ can be glued up along the open set corresponding to
$X_{\tau_{\alpha\beta}}$ which they have in common. By this process, the fan $\Sigma$ gives rise to an algebraic variety which is proper over ${\mathbf
A}^d(k)$
$$\pi (\Sigma)\colon Z(\Sigma)\to {\mathbf A}^d(k)$$
The variety $Z(\Sigma)$ is covered by affine charts corresponding to the $d$-dimensional cones $\sigma$ of $\Sigma$ and in each of these charts the
map $\pi(\Sigma)$ corresponds to the inclusion of algebras $k[u_1,\ldots ,u_d]\subset k[\check\sigma\cap M]$. If $\sigma$ is generated by $d$ vectors
with determinant $\pm 1$, i.e., which form a basis of the integral lattice, the second algebra is a polynomial ring $k[y_1,\ldots ,y_d]$ and the
inclusion is given by the monomial expression we started from. Let us give the following definition, and remember that $\{0\}$ is a face of each strictly convex
cone:
\begin{definition}\label{compat} A convex polyhedral cone $\sigma$ is compatible with a convex polyhedral cone $\sigma'$ if $\sigma\cap\sigma'$ is a face of
each. A fan is compatible with a polyhedral cone if each of its cones is.\end{definition} \begin{lemma} Given two monomials
$u^m,u^n$, if we can find a fan
$\Sigma$ which is
compatible with the hyperplane
$H_{m-n}$ in the weight space, then in each chart of $Z(\Sigma)$ the transform of one of our monomials will divide the other.\end{lemma}
\begin{proof} This follows from Lemma \ref{div}.\end{proof}
\par\medskip\noindent
Example: In dimension
$d=2$, let us seek to make one of the two monomials
$(u_1,u_2)$ divide the other after a monomial transformation. The hyperplane in the weight space is $w_1=w_2$; its intersection with the first quadrant
defines a fan whose cones are the two cones
$\sigma_1$ and
$\sigma_2$  generated by $a^1=(0,1),a^2=(1,1)$ and by $b^1=(1,1),b^2=(0,1)$ and their faces. The semigroup of integral points of $\check\sigma_1\cap
M$ is generated by $(1,0),(-1,1)$, which correspond respectively to the monomials $y_1=u_1,y_2=u_1^{-1}u_2$ and the semigroup of integral points of 
$\check\sigma_2\cap
M$ is generated by $(0,1),(1,-1)$, which correspond respectively to the monomials $y'_2=u_2,y'_1=u_1u_2^{-1}$. There is a natural isomorphism of the
open sets where $u_1\neq0,u_2\neq0$ and gluing up gives the two-dimensional subvariety of ${\mathbf A}^2(k)\times{\mathbf P}^1(k)$ defined by
$t_2u_1-t_1u_2=0$, where $(t_1:t_2)$ are the homogeneous coordinates on $\Pp^1(k)$, with its natural projection to ${\mathbf A}^2(k)$: it is the blowing-up of the
origin in ${\mathbf A}^2(k)$. 

\includegraphics[scale=0.4]{blowup.pdf}\par\medskip
Now if we have a finite number of distinct monomials $\neq 1$, say $u^{m^1},\ldots,u^{m^q}$, and if we can find a fan $\Sigma$ with support
$\check{\mathbf R}^d_{\geq 0}$ and compatible with \textit{all} the hyperplanes $H_{m^s-m^t},\ 1\leq s,t\leq q, s\neq t$, this will give us an algebraic (toric)
variety $Z(\Sigma)$, possibly singular and endowed with a \textit{proper surjective} map
$\pi(\Sigma)\colon
Z(\Sigma)\to {\mathbf A}^d(k)$ such that the pull back by the map $\pi(\Sigma)$ of the ideal ${\mathcal M}$ generated by our monomials is strongly principal in
each chart. The properness and surjectivity are ensured (see [TE]) by the fact that the support of $\Sigma$ is ${\mathbf R}_{\geq 0}^d$.\par
Our collection of hyperplanes $(H_{m^s-m^s},\ 1\leq s,t\leq q, ,s\neq t)$ through the origin in fact defines a fan $\Sigma_0(F)$ which depends only
upon the finite set
$F=\{m^1,\ldots ,m^q\}$ of elements of ${\mathbf Z}^d$: take as cones the closures of the connected components of the complement in
$\check{\mathbf R}^d_{\geq 0}$ of the union of all the hyperplanes. They are strictly convex rational cones because they lie in the first quadrant and are
bounded by hyperplanes whose equations have integral coefficients. Add all the faces of these cones, and we have a fan, of course not regular in general. To say
that a monomial ideal generated by monomials in the generators of the algebra $k[\check\sigma
\cap M]$ is locally strongly principal is not nearly as useful when these generators do not form
a system of coordinates as when they do. However, it is important to note that we first make our ideal
$\mathcal M$ locally strongly principal by the map $\pi (\Sigma_0)\colon Z (\Sigma_0)\to {\mathbf A}^d(k)$, and then resolve the singularities of $Z
(\Sigma_0)$ by a toric map.\par The second step corresponds to a refinement of $\Sigma_0$ into a regular fan $\Sigma$,  where refinement means that each
cone of the second fan in contained in a cone of the original.\Par This is always possible in view of the:\par 
\begin{theorem}\label{resev}{\rm (Kempf, Knudsen, Mumford, St. Donat, see [TE])}
\textit{ Given a rational fan, it is always possible to refine it into a regular fan.}\end{theorem}
From this follows:\par\medskip\noindent
\begin{theorem}\label{principalization} Let $k$ be a field. Given a monomial ideal ${\mathcal M}$ in $k[u_1,\ldots ,u_d]$, there exists a fan $\Sigma_0$ with
support
$\check{\mathbf R}^d_{\geq 0}$ such that, given any regular refinement $\Sigma$ of $\Sigma_0$, the associated proper birational toric map of non singular toric
varieties
$$\pi(\Sigma)\colon Z(\Sigma)\to {\mathbf A}^d(k)$$ has the property that the transform of ${\mathcal M}$ is strongly principal in each chart.\end{theorem}
\begin{remark} By construction, for each chart
$Z(\sigma)$ of $Z(\Sigma)$ there is an element of ${\mathcal M}$ whose transform generates the ideal ${\mathcal M}{\mathcal O}_{Z(\sigma)}$. This
element cannot be the same for all charts unless
${\mathcal M}$ is already principal. \end{remark} To see this, assume that there is a monomial $u^n$ whose transform generates ${\mathcal
M}{\mathcal O}_{Z(\Sigma)}$ in every chart. This means that every simplicial cone $\sigma$ of our fan with support $\check{\mathbf R}^d_{\geq 0}$ is in the
positive side of all the hyperplanes $H_{m-n}$ for all the other monomials
$u^m$ generating ${\mathcal M}$. But this is possible only if none of these hyperplanes meets the positive quadrant outside $\{0\}$, which means that
$u^n$ divides all the other $u^m$.
\begin{remark}  {\rm (Strong principalization and blowing-up)} Given a finitely generated ideal $I$ in a commutative integral domain $R$, there is a proper
birational map $\pi\colon B(I)\to \hbox{\rm Spec}R$, unique up to unique isomorphism, with the property that the ideal sheaf $I{\mathcal O}_{B(I)}$ generated
by the compositions with $\pi$ of the elements of $I$ is locally principal and generated by a non zero divisor (that is, an \textit{invertible} ideal), and that any map $W\to\hbox{\rm
Spec}R$ with the same property factors uniquely through $\pi$. The map $\pi$ is called the \textit{blowing-up} of $I$ in $R$, or in $\hbox{\rm Spec}R$.\par\noindent
The blowing-up is in particular independent of the choice of generators of $I$. Since a product of ideals is invertible if and only if each ideal is, for $I=(f_1,\ldots, f_s)R$,
the blowing-up in $\hbox{\rm Spec}R$ of the ideal $J=\prod_{i<j}(f_i,f_j)R$ will make $I$ strongly principal.
\par\noindent If
$I$ is a monomial ideal in
$k[u_1,\ldots ,u_d]$, according to {\rm [TE]}, the blowing-up of $I$ \textit{followed by normalization} is the equivariant map associated to the fan dual to the Newton
polyhedron of
$I$, two notions defined below respectively in section \ref{nondeg} and in the Appendix. The reader is encouraged to check that the fan just mentioned admits the
fan
$\Sigma_0$ introduced above as a refinement, illustrating the general fact that a strong principalization map factors through the blowing-up.\par\noindent
\rm{Strong principalization was stressed in these lectures because it is directly linked with the resolution of singularities of binomial ideals explained in section
\ref{resbin}.}
\end{remark}
\begin{exercise} Check that one can in all statements and proofs in this section replace the positive quadrant of $\check{\mathbf R}^d$ by any
strictly convex rational cone $\sigma_0\subset \check{\mathbf R}^d$. The affine space ${\mathbf A}^d(k)$ is then replaced by the affine toric variety
 $X_{\sigma_0}$. \end{exercise} 
\section{The integral closure of ideals}\label{intcl}
Given a finite set $F=\{m^1,\ldots ,m^q\}$ of elements of ${\mathbf Z}^d$, let us define its \textit{support function} as follows: it is the function
$\hbox{\rm h}_F\colon\check
{\mathbf R}^d\to {\mathbf R}$ defined by $$\hbox{\rm h}_F(\ell)=\hbox{\rm min}_{1\leq s\leq q}\ell (m^s).$$ 
For reasons which will become apparent, I denote the convex hull of $F$ by $\overline F$. It is a classical result on convexity,
presenting the convex hull of a set as the intersection of the half-spaces containing that set, that the convex hull of
$F$ is given by
$$\overline F=\{n\in {\mathbf R}^d/\ell (n)\geq \hbox{\rm h}_F(\ell)\ \ \forall \ell\in\check{\mathbf R}^d\}.$$
 It is often found in convexity books as saying that
a convex set is the intersection of the half-spaces determined by its support hyperplanes. The same proof shows that the ``positive convex hull'' is defined
by the same inequalities restricted to the linear forms which are in the positive quadrant of $\check{\mathbf R}^d$:
$$\overline{\bigcup_{1\leq s\leq q}(m^s+{\mathbf R}^d_{\geq 0})}=\{n\in {\mathbf R}^d/\ell (n)\geq \hbox{\rm h}_F(\ell)\ \ \forall \ell\in
\check{\mathbf R}_{\geq 0}^d\}.$$
\begin{lemma}\label{lin} The support function $\hbox{\rm h}_F$ is linear in each cone of the fan $\Sigma_0(F)$ introduced in section \ref{strongp}.\end{lemma}
\begin{proof} It follows directly from the definitions.\end{proof}
 Let us choose a strongly principalizing map $\pi(\Sigma)\colon Z(\Sigma)\to{\mathbf A}^d(k)$ with $\Sigma$ a refinement of $\Sigma_0(F)$ as in Theorem \ref
{principalization}. Then $Z(\Sigma)$ is normal by [TE] (it is regular if $\Sigma$ is regular), and $\pi(\Sigma)$ is proper and birational. Let $u^n$ be a monomial in
$k[u_1,\ldots ,u_d]$. Given a chart $X_\sigma$ of $Z(\Sigma )$, corresponding to $\sigma\in \Sigma$, a necessary and sufficient condition for
$u^n\circ\pi(\Sigma)$ to belong in $k[\check\sigma\cap M]$ to the ideal generated by the transforms of the generators of
${\mathcal M}$ is that we have $\ell (n)\geq \hbox{\rm h}_F(\ell)$ for all $\ell\in\sigma$: by Lemma \ref{lin}, we have for some $t\in\{1,\ldots ,q\}$
that
$\hbox{\rm h}_F(\ell)=\ell(m^t)$ for all $\ell\in\sigma$, and then by definition of $\check\sigma$ our inequality means that the quotient of the
transform of $u^n$ by the transform of $u^{m^t}$ is in $k[\check\sigma\cap M]$, which means that $u^nk[\check\sigma\cap M]\subset {\mathcal
M}k[\check\sigma\cap M]$. For this to be true in all charts it is necessary and sufficient, as we saw, that $n$ should be in the
convex hull $\overline{\bigcup_{1\leq s\leq q}(m^s+{\mathbf R}_{\geq 0}^d)}$. So we have finally, using a little sheaf-theoretic language (in
particular, $u^n{\mathcal O}_{Z(\Sigma)}=u^n\circ \pi(\Sigma)$ viewed as a global section of the sheaf ${\mathcal O}_{Z(\Sigma)}$):
\begin{lemma} We have
$u^n{\mathcal O}_{Z(\Sigma)}\in {\mathcal M}{\mathcal O}_{Z(\Sigma)}$ if and only if
$n$ is in the convex hull of
$\bigcup_{1\leq s\leq q}(m^s+{\mathbf R}_{\geq 0}^d)$.\end{lemma}
Now one defines integral dependance over an ideal (a concept which goes back to Pr\"ufer or even Dedekind) as follows:
\begin{definition} An element $h$ of a commutative ring $R$ is integral over an ideal $I$ of $R$ if it satisfies an algebraic relation
$$h^r+a_1h^{r-1}+\cdots +a_r=0\ \ \ \ \ \hbox{\rm with}\ \ a_i\in I^i\ \ \ \hbox{\rm for}\ 1\leq i\leq r .$$\end{definition}
It is not difficult to see that the set of elements integral over $I$ is an ideal $\overline I$ containing $I$ and contained in $\sqrt I$; it is the \textit{integral
closure} of $I$. We have the following characterization in algebraic geometry, which follows from the Riemann extension theorem:
\begin{proposition}{\rm (see [Li-T])} Let $k$ be a field and $R$ a localization of a finitely generated reduced $k$-algebra. Let $I$ be an ideal of $R$ and
$h\in R$. The element
$h$ is integral over $I$ if and only if there exists a proper and birational morphism $t\colon Z\to\hbox{\rm Spec}R$ such that $h\circ t\in I {\mathcal
O}_Z$ (i.e., $h{\mathcal O}_Z\in I {\mathcal O}_Z$), and then this inclusion holds for any such morphism such that
$Z$ is normal and $I{\mathcal O}_Z$ is invertible.\end{proposition}
From this follows the interpretation of the Lemma:
\begin{proposition} The integral closure in $k[u_1,\ldots ,u_d]$ of a monomial ideal generated by the monomials $u^{m^1},\ldots ,u^{m^q}$ is the monomial ideal generated by the
monomials with exponents in the convex hull $\overline E$ of
$E=\bigcup_{1\leq s\leq q}(m^s+{\mathbf R}_{\geq 0}^d)$. \end{proposition}
\begin{example} In the ring $k[u_1,\ldots ,u_d]$, for each integer $n\geq 1$ the integral closure of the ideal generated by $u_1^n,\ldots, u_d^n$ is
$(u_1,\ldots, u_d)^n$.\end{example}
\begin{exercise} Check that in the preceding subsection, one can in all statements and proofs replace the positive quadrant of ${\mathbf R}^d$ by any
strictly convex rational cone $\sigma_0\subset {\mathbf R}^d$ and let ${\mathcal M}$ denote the ideal generated by monomials
$u^{m^1},\ldots ,u^{m^q}$ of the normal toric algebra $k[\check{\sigma_0}\cap M]$; its integral closure $\overline{\mathcal M}$ in that algebra is generated
by the monomials with exponents in the convex hull in $\check{\sigma_0}$ of
$\bigcup_{1\leq s\leq q}(m^s+\check{\sigma_0})$.  \end{exercise} 
\includegraphics[scale=0.36]{essai1.pdf}\par\noindent
Note that the picture corresponds to a case where the generating system is not minimal.
\section{The monomial Brian\c con-Skoda Theorem}\label{BrSk}
\begin{theorem}{\rm{(Carath\'eodory's Theorem):}} Let $E$ be a subset of   ${\mathbf R}^d$; every point of the convex hull of $E$ is in the convex hull of $d+1$ points
of $E$.\end{theorem}
\begin{proof} For the reader's convenience, here is a sketch of the proof, according to [Gr\"u]. First one checks that the convex hull of $E$, defined as the
intersection of all convex subsets of ${\mathbf R}^d$ containing $E$, coincides with the set of points of ${\mathbf R}^d$ which are in the convex hull of a finite number of
points of $E$:\par\noindent Given a finite set $F$ of points of $E$, its convex hull $\overline F$ is contained in the convex hull $\overline E$ of $E$. Now  for two finite sets $F$ and
$F'$ we have
$\overline F\cup\overline F'\subseteq \overline{F\cup F'}$, so that $\bigcup_F\overline F$ is convex. It contains $E$ and so has to be equal to $\overline E$, which proves the
assertion. \par Given a point
$x$ of the convex hull of
$E$, let
$p$ be the smallest integer such that
$x$ is in the convex hull of
$p+1$ points of $E$, i.e., that $x=\sum_{i=0}^p\alpha_ix_i$, with $\alpha_i\geq 0, \ \sum_{i=0}^p\alpha_i=1$ and $x_i\in E$; we must prove that $p\leq d$. Assume that
$p>d$. Then the points $x_i$ must be affinely dependent: there is a relation $\sum_{i=0}^p\beta_ix_i=0$ with $\beta_i\in {\mathbf R}$, where not all the $\beta_i$
are zero and
$\sum_{i=0}^p\beta_i=0$. We may choose the $\beta_i$ so that at least one is $>0$ and renumber the points $x_i$ so that $\beta_p>0$ and for each index $i$
such that
$\beta_i> 0$ we have
$\frac{\alpha_i}{\beta_i}\geq\frac{\alpha_p}{\beta_p}$. For $0\leq i\leq p-1$ set $\gamma_i=\alpha_i-\frac{\alpha_p}{\beta_p}\beta_i$, and
$\gamma_p=0$. Now we have 
$$\sum_{i=0}^{p-1}\gamma_ix_i=\sum_{i=0}^p\gamma_ix_i=\sum_{i=0}^p\alpha_ix_i-\frac{\alpha_p}{\beta_p}\sum_{i=0}^p\beta_ix_i=x$$ and moreover
$$\sum_{i=0}^{p-1}\gamma_i=\sum_{i=0}^p\gamma_i=\sum_{i=0}^p\alpha_i-\frac{\alpha_p}{\beta_p}\sum_{i=0}^p\beta_i=1.$$
Finally each $\gamma_i$ is indeed $\geq 0$ since if $\beta_i\leq 0$ we have $\gamma_i\geq\alpha_i\geq 0$ and if $\beta_i>0$ and then by our choice of
numbering we have $\gamma_i=\beta_i(\frac{\alpha_i}{\beta_i}-\frac{\alpha_p}{\beta_p})\geq 0$. Assuming that $p>d$ we have expressed $x$ as the
barycenter of the $p$ points $x_0,\ldots ,x_{p-1}$ of $E$ with coefficients $\gamma_i$, which contradicts the definition of $p$ and thus proves the theorem.
\end{proof}
Taking for $E$ the set consisting of $d+1$ affinely independent points of ${\mathbf R}^d$ shows that the bound of the theorem is optimal. However, the following
result means that this is essentially the only case where $d+1$ points are necessary: 
\begin{proposition}\label{FHH} {\rm (Fenchel ([F]), Hanner-R\aa dstr\"om ([H-R]))} 
Let $E\subset {\mathbf R}^d$ be a subset having at most $d$ connected components. Every point of the convex hull of $E$
is in the convex hull of $d$ points of $E$.\end{proposition}
\begin{proof}(after [H-R]): Assume that a point $m$ of the convex hull is {\it not} in the convex hull of any subset of $d$ points
of $E$. By Caratheodory's theorem, $m$ is in the convex hull $\tau\subset {\bf R}^d$ of $d+1$
points of $E$; if these $d+1$ points were not linearly independent, the point $m$ would be in the convex hull of the intersection of $E$ with a hyperplane
and we could apply Caratheodory's theorem in a space of dimension
$d-1$ and contradict our assumption, so the convex hull $\tau$ of the $d+1$ points is a $d$-simplex. Let us choose
$m$ as origin, and denote by
$(q_0,\ldots ,q_d)$ the vertices of
$\tau$. We have therefore
$0=\sum_0^dr_iq_i$ with
$r_i\geq 0$ and $\sum_0^dr_i=1$. Our assumption that $0$ is not the barycenter of $d$ points implies that $0$ is in the interior of $\tau$, that is, $r_i>0$
for $0\leq i\leq d$. Consider the simplex
$-\tau$ and the cones with vertex
$0$ drawn on the faces of
$-\tau$. Since the $r_i$ are $>0$, we can reinterpret the expression of
$0$ as a barycenter of the $q_i$ to mean that each $q_i$ is in the cone with vertex $0$ generated by the vectors $-q_j$ for $j\neq i$; thus 
 each of these cones drawn from $0$ on the faces of $-\tau$ contains a point of $E$, namely one of the $q_i$. The union of their closures is ${\bf R}^d$
because
$-\tau$ is a $d$-simplex, and no point of $E$ can be on the boundary of one of these cones; if such was the case, this point,
together with $d-1$ of the vertices of $\tau$, would define a $(d-1)$-simplex with vertices in $E$ and containing $0$, a
possibility which we have excluded. Therefore these $d+1$ cones divide $E$ into $d+1$ disjoint non empty parts, and
$E$ must have at least $d+1$ connected components.\end{proof}
Now let us remark that, given finitely many points $m^1,\ldots ,m^q$ in the positive quadrant ${\mathbf R}_{\geq 0}^d$, the set $E=\bigcup_{s=1}^q(m^s+{\mathbf R}_{\geq
0}^d)$ is connected. Indeed, by definition, each point of this set is connected by a line to at least one of the points $m^s$, and any point of ${\mathbf R}_{\geq 0}^d$
having each of its coordinates larger than the maximum over $s\in\{1,\ldots, q\}$ of the corresponding coordinate of the $m^s$ is in $E$ and connected by
lines to all the points $m^s$, so that any two of the points $m^s$ are connected in $E$.\Par
Now let $\sigma$ be a strictly convex rational cone in $\check{\mathbf R}^d$ and $\check\sigma\subset {\mathbf R}^d$ its dual. We need not assume that $\sigma$ is
regular, or even simplicial. Let $m^1,\ldots ,m^q$ be integral points in $\check\sigma$, corresponding to monomials $u^{m^1},\ldots ,u^{m^q}$ in the
algebra $k[\check\sigma\cap M]$. The integral closure $\overline{\mathcal M}$ in $k[\check\sigma\cap M]$ of the ideal ${\mathcal M}$ generated by
the monomials $u^{m^s}$ is the ideal generated by the monomials $u^n$ such that $n$ is in the convex hull of the set
$E=\bigcup_{s=1}^q(m^s+\check\sigma)$. What we have just said about the connectedness of $E$ extends immediately.
\begin{theorem}{{\rm (Monomial Brian\c con-Skoda theorem)}} Let $k$ be a field and let $\sigma$ be a strictly convex rational cone in ${\mathbf R}^d$.
Given a monomial ideal
${\mathcal M}$ in
$k[\check\sigma\cap M]$, we have the inclusion of ideals
$$\overline{{\mathcal M}^d}\subset{\mathcal M}.$$
\end{theorem}
\begin{proof} (Compare with [T3]) Let $y_1,\ldots, y_N$ be a system of homogeneous generators of the graded $k$-algebra $k[\check\sigma\cap M]$ and let $y^{m^1},\ldots
,y^{m^q}$ be generators of
${\mathcal M}$ in
$k[\check\sigma\cap M]$. Set
$E=\bigcup_{1\leq s\leq q}(m^s+\check\sigma)\subset \check\sigma$. Thanks to Proposition
\ref{FHH} and the fact that $E$ is connected, it suffices to show that any point
$n\in
\check\sigma\cap M$ which is the barycenter of
$d$ points
$x_1,\ldots ,x_d$, each of which is the sum of $d$ points of $E$, is in $E$. But then $\frac{1}{d}n$ is also, as a barycenter of
barycenters of points of $E$, in the convex hull of $E$, and therefore, by Proposition \ref{FHH}, the barycenter of $d$ points of $E$. Let us write
$\frac{1}{d}n=\sum_{i=1}^dr_ie_i$ with $e_i\in E$,
$r_i\geq 0$ and $\sum_{i=1}^dr_i=1$. At least one of the $r_i$, say $r_1$, must be $\geq \frac{1}{d}$, so that $n\in e_1+\check\sigma\subset E$, which
proves the result. \end{proof}
\begin{exercise} Prove by the same method that for each integer $\lambda\geq 1$ we have $$\overline{{\mathcal M}^{d+\lambda-1}}\subset{\mathcal
M}^\lambda .$$\end{exercise}
\begin{remark} It is not difficult to check that $$\overline {E} =\lim_{n\to \infty}\frac{nE}{n}=\bigcup_{n\in {\mathbf N}}\frac{nE}{n},$$ where $nE$
means the Minkowski multiple, i.e., set of all sums of $n$ elements of $E$, while dividing by $n$ meant homothety of ratio $\frac{1}{n}$. In fact, the inclusion
$\bigcup_{n\in
{\mathbf N}}\frac{nE}{n}\subset \overline E$ is clear, and the first set is also clearly convex, so they are equal. The combinatorial avatar of the weak form of
the Brian\c con-Skoda theorem which states that
$x\in\overline{\mathcal M}\implies x^d\in{\mathcal M}$  is the existence of a uniform bound for the $n$ such that $x\in\overline E\implies nx\in
E$, namely
$n=d$. \par\noindent
\end{remark}
The Brian\c con-Skoda theorem is the statement $\overline{{\mathcal M}^d}\subset {\mathcal M}$ for an ideal in a $d$-dimensional regular local ring. The
rings $k[\check\sigma\cap M]$ are not regular in general, nor are they local, but the monomial Brian\c con-Skoda theorem for ideals in their localizations 
$k[\check\sigma\cap M]_m$ follows from the results of [Li-T] in the case where
${\mathcal M}$ contains an ideal generated by a regular sequence and with the same integral closure, since $k[\check\sigma\cap M]$ has only rational
singularities (see [TE]) and hence $k[\check\sigma\cap M]_m$ is a pseudo-rational local ring.\par\medskip The Brian\c con-Skoda theorem was originally
proved (see [B-S]) by analytic methods for ideals of
${\mathbf C}\{z_1,\ldots ,z_d\}$ and has been the subject of many algebraic proofs and generalizations. The first algebraic proof was given in [Li-T], albeit
for a restricted class of ideals in an extended class of rings (pseudo-rational ones). I refer the reader to Hochster's and Lazarsfeld's lectures in this volume for
references and recent developments.
\section{Polynomial ideals and non degeneracy}\label{nondeg}
The hypothesis of non degeneracy of a polynomial with respect to its Newton polyhedron has a fairly ancient history in the sense that it was made more or less implicitely by authors trying to compute various invariants of a projective hypersurface from its Newton polyhedron.  In the 19$^{th}$ Century one may quote Minding and Elliott, and in the twentieth Baker and Hodge (see \cite{Hod}). The last three were interested in computing the geometric genus of a projective curve or surface with isolated singularities from its Newton polygon or polyhedron. This is a special case of computation of a multiplier
ideal (see [M-T], and compare its Theorem of Hodge 2.3.1 with the recent work of J. Howald expounded in Lazarsfeld's lectures in this volume; see also [Ho]).\par
The modern approach to non degeneracy was initiated essentially by Kushnirenko and Khovanskii who made the non-degeneracy condition explicit and computed
from the Newton polyhedron invariants of a similar nature. In particular Khovanskii gave the general form of Hodge's result. The essential facts behind the
classical computations turned out to be that non degenerate singularities have embedded toric (pseudo-)resolutions which depend only on their Newton
polyhedron and from which one can read combinatorially various interesting invariants, and that in the spaces of coefficients of all those functions or systems of
functions having given polyhedra, those which are non-degenerate are Zariski-dense.\par\medskip\noindent
 Let
$$f=\sum_pf_pu^p$$ be an arbitrary polynomial or power series in
$d$ variables with coefficients in the field
$k$. Let
$\hbox{\rm Supp}(f)=\{p\in {\mathbf R}^d/f_p\neq 0\}$ be its \textit {support}. The affine Newton polyhedron of $f$ in the coordinates $(u_1,\ldots ,u_d)$ is the
boundary ${\mathcal N}(f)$ of the convex hull in
${\mathbf R}^d_{\geq 0}$  of the support of $f$. The local Newton polyhedron is the boundary ${\mathcal N}_+(f)$ of $${\mathcal P}_+(f)=\hbox{\rm convex hull of}\
(\hbox{\rm Supp}(f)+{\mathbf R}^d_{\geq 0}).$$
It has finitely many compact faces and its non compact faces of dimension $\leq d-1$ are parallel to coordinate planes (intersections of coordinate hyperplanes). Note that both
polyhedra depend not only on $f$ but also on the choice of coordinates. Remark also that the local Newton polyhedron has no interest if the
polynomial or series $f$ has a non zero constant term.\par We can define the affine (resp. local) support function associated with the function $f$ as
follows (in the affine case, it is the same definition as that given above, applied to the set of monomials appearing in $f$): .
\par\noindent For the affine Newton polyhedron it is the function defined on $\check{\mathbf R}^d$ by
$$\hbox{\rm h}_{\mathcal N (f)}(\ell)=\hbox{\rm min}_{p\in \mathcal N (f)}\ell(p),$$
and for the local Newton polyhedron it is defined on the first quadrant $\check{\mathbf R}_{\geq 0}^d$ by
$$\hbox{\rm h}_{\mathcal N_+ (f)}(\ell)=\hbox{\rm min}_{p\in \mathcal N_+ (f)}\ell(p).$$
Both functions are piecewise linear in their domain of definition, which means that there is a decomposition of their respective domains of definition into
convex cones such that the corresponding functions are linear in each cone. These collections of cones are actually fans, in $\check {\mathbf R}^d$ and $\check
{\mathbf R}^d_{\geq 0}$ respectively. These fans are ''dual'' to the Newton polyhedra in the following sense:\par\noindent
Consider, say for the local polyhedron, the following equivalence relation between linear functions:
$$\ell\equiv \ell'\iff \{p\in \mathcal N_+ (f)\vert \ell (p)=\hbox{\rm h}_{\mathcal N_+ (f)}(\ell)\}=\{p\in \mathcal N_+ (f)\vert \ell' (p)=\hbox{\rm
h}_{\mathcal N_+ (f)}(\ell')\}$$
Then, the equivalence classes form a decomposition of the first quadrant into strictly convex rational cones, and by definition the support function is linear
in each of them, given by $\ell\mapsto \ell (p)$ for any $p$ in the set $\{p\in \mathcal N_+ (f)\vert \ell (p)=\hbox{\rm h}_{\mathcal N_+ (f)}(\ell)\}$. These
sets are {\it faces} of the Newton polyhedron, and the
collection of the cones constitutes a fan $\Sigma_{\cal N}$ in ${\mathbf R}^d_{\geq 0}$, called the \textit{dual fan} of the Newton polyhedron. This establishes a
\textit{one to one decreasing correspondence between the cones (except for the origin) of the dual fan of a Newton polyhedron and the faces of all dimensions of that Newton
polyhedron}. Note that the cones corresponding to non compact faces of the Newton polyhedron meet coordinate hyperplanes outside the origin.\par
 \par\medskip\noindent
We have now associated to each polynomial $f=\sum_pf_pu^p$ a dual fan in $\check{\mathbf R}^d$ corresponding to the global Newton polyhedron, and another
one in
$\check{\mathbf R}^d_{\geq 0}$ corresponding to the local Newton polyhedron. The local polyhedron is also defined for a series $f=\sum_pf_pu^p$, and the
combinatorial constructions are the same. Let us for the moment restrict our attention to the local polyhedron, assuming
that $f_0=0$ and let us choose a regular refinement $\Sigma$ of the fan associated to it.\par\noindent 
By the definition just given, this means that for each cone $\sigma=\langle a^1,\ldots ,a^k\rangle$ of the fan $\Sigma$, the primitive vectors $a^i$ form
part of a basis of the integral lattice, and all the linear forms $p\mapsto \langle a^i,p\rangle$, when restricted to the set $\{p/f_p\neq 0\}$, take their
minimum value on the same subset, which is a face, of the (local) Newton polyhedron of $f=\sum_pf_pu^p$. This face may or may not be compact.\par
Let us examine the behavior of
$f$ under the map
$\pi (\sigma)\colon Z(\sigma)\to
{\mathbf A}^d(k)$ corresponding to a cone
$\sigma=\langle a^1,\ldots ,a^d\rangle\subset \check{\mathbf R}^d_{\geq 0}$ of a regular fan which is a subdivision of the fan associated to the local polyhedron of
$f$. We assume that $f$ is not divisible by any of the coordinates. If we write
$\hbox{\rm h}$ for
$\hbox{\rm h}_{\mathcal N_+(f)}$ we get 
$$\begin{array}{lr}f\circ\pi(\sigma)=\cr \sum_pf_py_1^{\langle a^1,p\rangle}\ldots y_d^{\langle a^d,p\rangle}=y_1^{h(a^1)}\ldots
y_d^{h(a^d)}\sum_pf_py_1^{\langle a^1,p\rangle- h(a^1)}\ldots y_d^{\langle
a^d,p\rangle -h(a^d)}\end{array}$$
The last sum is by definition the \textit{strict transform} of $f$ by $\pi (\sigma)$.\par
I leave it as an exercise to check that:\par\noindent
a) In each chart $Z(\sigma)$ the exceptional divisor consists (set-theoretically) of the union of those hyperplanes $y_j=0$ such that $a^j$ is not a basis
vector of $\check {\mathbf Z}^d$.\par\noindent
b) Provided that no monomial in the $u_i$ divides $f$, the hypersurface $$\sum_pf_py_1^{\langle a^1,p\rangle- h(a^1)}\ldots y_d^{\langle
a^d,p\rangle -h(a^d)}=0$$ is indeed the strict transform by the map
$\pi (\sigma)\colon Z(\sigma)\to {\mathbf A}^d(k)$ of the hypersurface $X\subset {\mathbf A}^d(k)$ defined by $f(u_1,\ldots ,u_d)=0$, in the sense that it is the closure in
$Z(\sigma)$ of the image of $X\cap (k^*)^d$ by the isomorphism induced by $\pi (\sigma)$ on the tori of the two toric varieties  $Z(\sigma)$ and
${\mathbf A}^d(k)$ as well as in the sense that it is obtained from $f\circ\pi(\sigma)$ by factoring out as many times as possible the defining functions of the
components of the exceptional divisor.
\par\medskip\noindent Let us denote by
$\tilde f$ the strict transform of $f$ and note that by construction it has a non zero constant term: the cone $\sigma$ is of maximal dimension, which means
that there is a unique exponent
$p$ such that
$\langle a,p\rangle=h(a)$ for
$a\in\sigma$. If we consider the map $\pi (\tau)$ associated to a face $\tau$ of $\sigma$, it coincides with the restriction of $\pi (\sigma)$ to an
open set $Z(\tau)\subset Z(\sigma)$ which is of the form $y_j\neq 0$ for $j\in J$ where $J$ depends on $\tau\subset \sigma$.
 \par Now we can, for each cone
$\sigma$ of our regular fan, stratify the space
$Z(\sigma)$ in such a way that
$\pi (\sigma)^{-1}(0)$ is a union of strata. Let
$I$ be a subset of $\{1,2,\ldots ,d\}$ and define $S_I$ to be the constructible subset of $Z(\sigma)$ defined by $y_i=0\ \hbox{\rm for}\ i\in I,\
y_i\neq 0\ \hbox{\rm for}\ i\notin I$. The $S_I$ for $I\subset\{1,2,\ldots ,d\}$ constitute a partition of $Z(\sigma)$ into non singular varieties,
constructible in $Z(\sigma)$, which we call the \textit{natural stratification} of  $Z(\sigma)$. Note that if we glue up two charts $Z(\sigma)$ and $Z(\sigma')$ along
$Z(\sigma \cap\sigma')$, the natural stratifications glue up as well.\par\noindent If we restrict the strict transform
$$\tilde f(y_1,\ldots ,y_d)=\sum_pf_py_1^{\langle a^1,p\rangle- h(a^1)}\ldots y_d^{\langle
a^d,p\rangle -h(a^d)}$$
to a stratum $S_I$, we see that in the sum representing $\tilde f(y_1,\ldots ,y_d)$ only the terms $f_py_1^{\langle a^1,p\rangle- h(a^1)}\ldots
y_d^{\langle a^d,p\rangle -h(a^d)}$ such that $\langle a^i,p\rangle- h(a^i)=0$ for $i\in I$ survive. These equalities define a unique face $\gamma_I$ of the
Newton polyhedron of $f$, since our fan is a subdivision of its dual fan. Given a series $f=\sum_pf_pu^p$ and a weight vector $\check a\in {\mathbf R}_{\geq
0}^d$, the set $\{p\in {\mathbf Z}_{\geq 0}^d/f_p\neq 0\ \hbox{\rm and }\ \langle
\check a,p\rangle =h(\check a)\}$ is a face of the local Newton polyhedron of $f$, corresponding to the cone of the dual fan which contains $\check a$ in its
relative interior. If all the coordinates of the vector
$\check a$ are positive, this face is compact.\par Moreover, if we define
$$f_{\gamma_I}=\sum_{p\in\gamma_I}f_pu^p$$
to be the partial polynomial associated to the face $\gamma_I$, which is nothing but the sum of the terms of $f$ whose exponent is in the
face $\gamma_I$, we see that we have the fundamental equality
$$\tilde f\vert_{S_I}=\widetilde{f_{\gamma_I}}\vert_{S_I}$$
and we remark moreover that $\widetilde{f_{\gamma_I}}$ is a function on $Z(\sigma)$ which is independent of the coordinates $y_i$ for $i\in I$, so that it
is determined by its restriction to $S_I$.\par Now, to say that the strict transform $\tilde f=0$ in $Z(\sigma)$ of the hypersurface $f=0$ is transversal to the
stratum $S_I$ and is non singular in a neighborhood of its intersection with it is equivalent to saying that the restriction $\tilde f\vert_{S_I}$ of the
function $\tilde f$ defines,  by the equation $\tilde f\vert_{S_I}=0$, a non singular hypersurface of $S_I$. By the definition of
$S_I$ and what we have just seen, this in turn is equivalent to saying that the equation $\widetilde{f_{\gamma_I}}=0$ defines a non singular hypersurface
in the torus
$(k^*)^d=\{u/\prod_1^du_j\neq 0\}$ of
$Z(\sigma)$, and this finally is equivalent to saying that $ f_{\gamma_I}=0$ defines a non singular hypersurface in the torus $(k^*)^d$ of the affine space
${\mathbf A}^d(k)$ since
$\pi(\sigma)$ induces an isomorphism of the two tori. \par
This motivates the definition:
\begin{definition} The series $f=\sum_pf_pu^p$ is \textit{ non degenerate} with respect to its Newton polyhedron in the coordinates
$(u_1,\ldots ,u_d)$ if for every compact face $\gamma$ of ${\mathcal N}_+(f)$ the polynomial $f_\gamma$ defines a non singular hypersurface
of the torus $(k^*)^d$.\end{definition}
\begin{remark} By definition of the faces of the Newton polyhedron and of the dual fan, in each chart $Z(\sigma)$ of a regular fan refining the dual fan of ${\mathcal
N}_+(f)$, the compact faces $\gamma_I$ correspond to strata
$S_I$ of the canonical stratification which are contained in
$\pi(\sigma)^{-1}(0)$. Each stratum $S_I$ which is not contained in $\pi(\sigma)^{-1}(0)$ contains in its closure strata which are. \end{remark}
\begin{proposition}\label{reshyp}
If the germ of hypersurface $X$ is defined by the vanishing of a series $f$ which is non-degenerate, there is a neighborhood $U$ of $0$ in ${\mathbf A}^d(k)$ (a formal neighborhood if
the series $f$ does not converge) such that the strict transform of $X\cap U$ by the toric map $\pi(\Sigma)\colon Z(\Sigma)\to {\mathbf A}^d(k)$
associated to a regular fan refining the dual fan of its Newton polyhedron is non singular and transversal in each chart to the strata of the canonical stratification.\end{proposition} \begin{proof} By the fundamental equality
seen above, the restriction of the strict transform to one of the strata contained in
$\pi(\sigma)^{-1}(0)$, say $S_I$, has the same behavior as the
restriction of
$f_{\gamma_I}$, where
$\gamma_I$ is a compact face of the Newton polyhedron of $f$, to the torus
$(k^*)^d$. As we saw, this implies that the strict transform of $X\cap U$ is non singular and transversal to $S_I$. By openness of transversality the same transversality holds, whithin a
neighborhood of each point of  $\pi(\Sigma)^{-1}(0)$, for all strata.\par\noindent  Since the map
$\pi(\Sigma)\colon Z(\Sigma)\to {\mathbf A}^d(k)$ is proper, there is a neighborhood
$U$ of $0$ in ${\mathbf A}^d(k)$ such that the strict transform by $\pi (\Sigma)$ of the hypersurface $X\subset{\mathbf A}^d(k)$ is non singular in $\pi
(\Sigma)^{-1}(U)$ and transversal in each chart $Z(\sigma)$ to all the strata of the canonical stratification.\end{proof}\par\medskip The definition and properties of non
degeneracy extend to systems of functions as follows:\par Let $f_1,\ldots ,f_k$ be series in the variables $u_1,\ldots ,u_d$ defining a subspace $X\subset {\mathbf
A}^d(k)$ in a neighborhood of
$0$. For each
$j=1,\ldots ,k$ we have a local Newton polyhedron ${\mathcal N}_+(f_j)$. Let us choose a regular fan $\Sigma$ of ${\mathbf R}_{\geq 0}^d$ compatible with all
the fans dual to the polyhedra ${\mathcal N}_+(f_j)$ for $j=1,\ldots ,k$. We have for each $j$ the same correspondence as above between the strata $S_I$ of each
chart
$Z(\sigma)$ for $\sigma\in \Sigma$ and the faces of ${\mathcal N}_+(f_j)$, the strata contained in $\pi(\sigma)^{-1}(0)$ corresponding to compact faces.\par
For each vector $\check a\in \check {\mathbf R}_{\geq 0}^d$ we get as above a system of equations $f_{1,\check a},\ldots ,f_{k,\check a}$ where $f_{j,\check a}=\sum_{\{p/\langle
\check a,p\rangle=h(\check a)\}}f_{jp}u^p$. 
\begin{definition} The system of equations $f_1,\ldots ,f_k$ is said to be non degenerate of rank $c$ with respect to the Newton polyhedra of the $f_j$ in the
coordinates $(u_1,\ldots ,u_d)$ if no coordinate is a zero divisor modulo the ideal generated by $f_1,\ldots ,f_k$ and for each vector
$\check a\in {\mathbf R}_{>0}^d$ the ideal of
$k[u_1^{\pm1},\ldots ,u_d^{\pm1}]$ generated by the polynomials $f_{1,\check a},\ldots ,f_{k,\check a}$ defines a non singular subvariety of dimension $d-c$ of
the torus
$(k^*)^d$. 
\end{definition}
\begin{remark} The non degeneracy condition does not in general imply that for
each choice of a compact face $\gamma_j$ in each ${\mathcal N}_+(f_j)$, the ideal generated by the polynomials $f_{1\gamma_1},\ldots ,f_{k\gamma_k}$
defines a non singular subvariety of dimension $d-c$ of the torus $(k^*)^d$. One can interpret this in terms of the positions of the strict transforms of the $f_j$ after a desingularizing toric map as in the next proposition.\end{remark}
Exactly as in the case of hypersurfaces, one then has the
\begin{proposition} If the system of equations $f_1,\ldots ,f_k$ is non degenerate of rank $c$, for any regular fan $\Sigma$ of ${\mathbf R}_{\geq 0}^d$
compatible with the dual fans of the polyhedra ${\mathcal N}_+(f_j)$, there is a neighborhood $U$ of $0$ in ${\mathbf A}^d(k)$ (a formal neighborhood if
all the series $f_j$ do not converge) such that the strict transform
$X'\subset Z(\Sigma)$ by the toric map $\pi(\Sigma)\colon Z(\Sigma)\to {\mathbf A}^d(k)$ of the subvariety $X\cap U$ defined in $U$ by the ideal generated by
$f_1,\ldots ,f_k$ is non singular and of dimension $d-c$ and transversal to the strata of the natural stratification in $\pi(\Sigma)^{-1}(U)$.\end{proposition}
The proof is the same as that of Proposition \ref{reshyp}.\par\medskip
There is a difference, however, between the birational map $X'\to X\cap U$ induced by $\pi (\Sigma)$ and a resolution of singularities; this map is not necessarily
an isomorphism outside of the singular locus; it is therefore only a \textit{pseudo-resolution} in the sense of [G-T]. In fact, even in the non degenerate case,
and even for a hypersurface, the Newton polyhedron contains in general far too little information about the singular locus of $X$ near $0$. Kushnirenko
introduced, for isolated hypersurface singularities, the notion of being
\textit{convenient} with respect to a coordinate system. It means that the Newton polyhedron meets all the coordinate axis of ${\mathbf R}_{\geq 0}^d$. For a
hypersurface with isolated singularity, it implies that a toric pseudo-resolution associated to the Newton polyhedron is a resolution. This was extended and
generalized by M. Oka for complete intersections. The reader is referred to ([Ok], Chap. III, especially Theorem 3.4) and we will only quote here the following
fact which is also a consequence of the existence of a toric pseudo-resolution (see [Ok], Chap. III, Lemma 2.2):\par\noindent \textit{If $k$ is a field and
$(X,0)\subset {\mathbf A}^d(k)$ is a germ of a complete intersection with equations $f_1=\ldots =f_c=0$, which is non degenerate with respect to the Newton
polyhedra of its equations in the coordinates $u_1,\ldots, u_d$, then there is a (possibly formal) neighborhood $U$ of $0\in {\mathbf A}^d(k)$ such that the
intersection of
$X$ and the torus
$(k^*)^d$ has no singularities in
$U$.}\par\noindent In the formal case this should be understood as saying that formal germ at $0$ of the singular locus of $X$ is contained in the union of the
coordinate hyperplanes. 
\par\medskip\noindent
Finally, it seems that the following coordinate-free definition of non degeneracy is appropriate:
\begin{definition} An algebraic or formal subscheme $X$ of an affine space ${\mathbf A}^d(k)$ is non degenerate at a point $x\in X$ if there
exist local coordinates $u_1,\ldots ,u_d$ centered at $x$ and an open (\'etale or formal) neighborhood $U$ of $x$ in ${\mathbf A}^d(k)$ such that there is a
proper birational toric map
$\pi\colon Z\to U$ in the coordinates $u_1,\ldots ,u_d$ with $Z$ non singular and such that the strict transform $X'$ of $X\cap U$ by $\pi$ is non
singular and transversal to the exceptional divisor at every point of
$\pi^{-1}(x)\cap X'$.
\end{definition}\noindent
If $X$ admits a system of equations which in some coordinates is non degenerate with respect to its Newton polyhedra, it is also non degenerate in this sense as we
saw. The converse will not be discussed here.\par\noindent
 Then one can state the: \par\medskip\noindent
\textbf {Question} (see [T2]) Given a reduced and equidimensional algebraic or formal space $X$ over an algebraically closed field $k$, is it true that for
every point $x\in X$ there is a local formal embedding of $X$ into an affine space ${\mathbf A}^N(k)$ such that $X$ is non degenerate in ${\mathbf
A}^N(k)$ at the point $x$?
\par\noindent
A subsequent problem is to give a geometric interpretation of the systems of coordinates in which an embedded toric resolution for $X$
exists.\par\medskip\noindent
For branches (analytically irreducible curve singularities), the question is answered positively, and the problem settled in section \ref{resb} below. Recent
work of P. Gonz\'alez P\'erez ([GP]) also settles question and problem for irreducible quasi-ordinary hypersurface singularities.\par
In [T2] one finds an approach to the simpler problem where the non degeneracy is requested only with respect to a valuation of the local ring of $X$ at $x$. 
\par\medskip\noindent In a given coordinate system, and for given Newton polyhedra, ``almost all'' systems of polynomials having these given Newton
polyhedra are non degenerate with respect to them. In this sense there are many non degenerate singularities. However, the non degenerate singularities are
very special from the viewpoint of the classification of singularities. A plane complex branch is non degenerate in some coordinate system if and only if it has
only one characteristic pair, which means that its equation can be written in some coordinate system
$$u_1^p-u_0^q+\sum_{\frac{i}{q}+\frac{j}{p}>1}a_{ij}u_0^iu_1^j=0,$$ with $a_{ij}\in k$, and the integers $p,q$ are coprime.
The curve $$(u_1^2-u_0^3)^2-u_0^5u_1=0$$ is degenerate in any coordinate system since it has two characteristic pairs (see [Sm], [B-K]) .
\section{Resolution of binomial varieties}\label{resbin}
This section presents what is in a sense the simplest class of non degenerate singularities, according to the results in [GP-T]:\par Let
$k$ be a field. Binomial varieties over
$k$ are irreducible varieties of the affine space
${\mathbf A}^d(k)$ which can, in a suitable coordinate system, be defined by the vanishing of binomials in these coordinates i.e.,
expressions of the form $u^m-\lambda_{mn}u^n$ with $\lambda_{mn}\in k^*$.  An ideal generated by such binomial expressions is called a \textit{binomial ideal}.
These affine varieties defined by prime binomial ideals are also (see [St]) the irreducible affine varieties on which a torus of the same dimension acts algebraically with a dense orbit; they
are the (not necessarily normal) \textit{affine toric varieties}.\par Binomial ideals were studied in [E-S]; Eisenbud and Sturmfels showed in particular that if $k$ is algebraically closed
their geometry is determined by the lattice generated by the differences $m-n$ of the exponents of the generating binomials. If the field
$k$ is not algebraically closed, the study becomes more complicated. Here I will assume throughout that $k$ is algebraically closed. It is natural to study the
behavior of binomial ideals under toric maps.\par Let
$\sigma=\langle a^1,\ldots ,a^d\rangle$ be a regular cone in $\check {\mathbf R}^d_{\geq 0}$. The image of a binomial $u^m-\lambda_{mn}u^n$ by the map
$k[u_1,\ldots ,u_d]\to k[y_1,\ldots ,y_d]$ determined by $u_i\mapsto y_1^{a^1_i}\ldots y_d^{a^d_i}$ is given by: $$u^m-\lambda_{mn}u^n\mapsto y_1^{\langle
a^1,m\rangle}\ldots y_d^{\langle a^d,m\rangle}-\lambda_{mn}y_1^{\langle a^1,n\rangle}\ldots y_d^{\langle a^d,n\rangle}.$$ In general this only tells us that
the transform of a binomial is a binomial, which is no news since by definition of a toric map the transform of a monomial is a monomial.
\par\noindent However, something interesting happens if we assume that the cone $\sigma$ is \textit{compatible} with the hyperplane $H_{m-n}$ which is the
dual in the space of weights of the vector $m-n$ of the space of exponents, in the sense of definition \ref{compat}, where we remember that the origin $\{0\}$ is
a face of any polyhedral cone. Note that the Newton polyhedron of a binomial has only one compact face, which is a segment, so that for a cone in $\check
{\mathbf R}^d_{\geq 0}$, being compatible with the hyperplane $H_{m-n}$ is the same as being compatible with the dual fan of the Newton polyhedron of our
binomial.\par Let us assume that the binomial hypersurface
$u^m-\lambda_{mn}u^n=0$ is irreducible; this means that no variable
$u_j$ appears in both monomials, and the vector $m-n$ is primitive. In the sequel, I will tacitly assume this and also that our binomial is really singular, i.e., not 1of the form
$u_1-\lambda u^m$.\par\noindent  If the convex cone
$\sigma$ of dimension $d$ is compatible with the hyperplane
$H_{m-n}$, it is contained in one of the closed half-spaces determined by
$H_{m-n}$. This means that all the non zero $\langle a^i,m-n\rangle$ have the same sign, say positive. It also means that, if we renumber the vectors $a^i$
in such a way that $\langle a^i,m-n\rangle=0$ for $1\leq i\leq t$ and $\langle a^i,m-n\rangle>0$ for $t+1\leq i\leq d$, we can write the transform of our
binomial in the following way:
$$u^m-\lambda_{mn}u^n\mapsto y_1^{\langle a^1,n\rangle}\ldots
y_d^{\langle a^d,n\rangle}(y_{t+1}^{\langle a^{t+1},m-n\rangle}\ldots
y_d^{\langle a^d,m-n\rangle}-\lambda_{mn}).$$
And we can see that the strict transform $y_{t+1}^{\langle a^{t+1},m-n\rangle}\ldots
y_d^{\langle a^d,m-n\rangle}-\lambda_{mn}=0$ of our hypersurface in the chart $Z(\sigma)$ is non singular!\par\noindent It is also irreducible in view of the results of [E-S] because
we assumed that the vector
$m-n$ is primitive and the matrix $(a^i_j)$ is unimodular. This implies that the vector $(0,\ldots, 0,\langle a^{t+1},m-n\rangle,\ldots ,\langle a^d,m-n\rangle)$ is also primitive, and
the strict transform irreducible. Moreover, in the chart
$Z(\sigma)$ with
$\sigma=\langle a^1,\ldots ,a^d\rangle$, the strict transform meets the hyperplane
$y_j=0$ if and only if 
$\langle a^j,m-n\rangle=0$. Unless our binomial is non singular, a case we excluded, this implies that $a^j$ is not a vector of the canonical basis of $W$, so that $y_j=0$ is a
component of the exceptional divisor. So we see that the strict transform meets the exceptional divisor only in those charts such that $\sigma\cap H_{m-n}\neq\{0\}$,
and then meets it transversally. \par
So we have in this very special case achieved that the total transform of our irreducible binomial hypersurface defines in each chart a \textit{divisor with normal crossings} that is, a
divisor locally at every point defined in suitable local coordinates by the vanishing of a monomial and whose irreducible components are non singular.
\par Now let us consider a prime binomial ideal of $k[u_1,\ldots ,u_d]$ generated by $(u^{m^\ell}-\lambda_\ell u^{n^\ell})_{\ell \in \{1,\ldots , L\}},\
\lambda_\ell\in k^*$. Let us denote by $\mathcal L$ the sublattice of ${\mathbf Z}^d$ generated by the differences $m^\ell-n^\ell$. According to [E-S], the
dimension of the subvariety $X\subset {\mathbf A}^d(k)$ defined by the ideal is $d-r$ where $r$ is the rank of the ${\mathbf Q}$-vector space $\mathcal L\otimes_{\mathbf
Z}{\mathbf Q}$. To each binomial is associated a hyperplane $H_\ell\subset \check{\mathbf R}^d$, the dual of the vector $m^\ell-n^\ell\in {\mathbf R}^d$. The intersection
${\mathcal W}$ of the hyperplanes
$H_\ell$ is the dual of the vector subspace ${\mathcal L}\otimes_{\mathbf Z}{\mathbf R}$ of ${\mathbf R}^d$ generated by the vectors $m^\ell-n^\ell$; its dimension is $d-r$.
\par\noindent Let $\Sigma$ be a fan with support ${\mathbf R}^d_{\geq 0}$ which is compatible with each of the\break hyperplanes $H_\ell$. 
Let us compute the transforms of the generators $u^{m^\ell}-\lambda_\ell u^{n^\ell}$ in a chart $Z(\sigma)$ associated to the cone $\sigma=\langle a^1,\ldots
,a^d\rangle$: after renumbering the vectors $a^j$ and possibly exchanging some $m^\ell,n^\ell$ and replacing $\lambda_\ell$ by its inverse, we may assume
that
$a^1,\ldots ,a^t$ are in
${\mathcal W}$, that the $\langle a^j,m^\ell-n^\ell\rangle$ are $\geq 0$ for $j=t+1,\ldots ,d$, and that moreover for each such index $j$ there is an $\ell$ such $\langle
a^j,m^\ell-n^\ell\rangle>0$. The transforms of the binomials can be written
$$y_1^{\langle a^1,n^\ell\rangle}\ldots y_d^{\langle a^d,n^\ell\rangle}(y_{t+1}^{\langle a^{t+1},m^\ell-n^\ell\rangle}\ldots y_d^{\langle
a^d,m^\ell-n^\ell\rangle}-\lambda_\ell)$$ with that additional condition.
If $\sigma\cap {\mathcal W}=\{0\}$, we have $t=0$ and the subvariety defined by the equations just written (the strict transform of $X$ in $Z(\sigma)$) does not meet
any coordinate hyperplane; in particular it does not meet the exceptional divisor. In general, still assuming that none of the binomials is already in the form
$u_j-\lambda$, one sees that the additional condition implies that, just like in the case of hypersurfaces, the strict transform meets the hyperplane $y_j=0$ if and only if $a^j$ is in
${\mathcal W}$.\par Now the claim is that in each chart $Z(\sigma)$ the strict transform is either empty or non singular and transversal to the exceptional divisor.\par
The ${\mathbf Q}$-vector space generated by the $m^\ell-n^\ell$ is of dimension $r$. Let us assume that $m^1-n^1,\ldots ,m^r-n^r$ generate it and let us denote by
$\mathcal L_1$ the lattice which they generate in ${\mathbf Z}^d$. By construction, the quotient ${\mathcal L}/{\mathcal L_1}$ is a torsion ${\mathbf Z}$-module. Let us first
show that the strict transform of the subspace $X_1\subset X$ defined by the first $r$ binomial equations is non singular and transversal to the exceptional
divisor.\par
We consider then, for each cone $\sigma=\langle a^1,\ldots ,a^d\rangle$, the equations 
$$\begin{array}{ccl}y_1^{\langle a^1,m^1-n^1\rangle}\cdots y_d^{\langle
a^d,m^1-n^1\rangle}-\lambda_1&=&0\\ y_1^{\langle a^1,m^2-n^2\rangle}\cdots y_d^{\langle
a^d,m^2-n^2\rangle}-\lambda_2&=&0\\ \vdots&\vdots&\vdots\\ y_1^{\langle
a^1,m^r-n^r\rangle}\cdots y_d^{\langle a^d,m^r-n^r\rangle}-\lambda_r&=&0
\end{array}$$ of the strict transform of $X_1$ in $Z(\sigma)$.\par\noindent 
We can compute by logarithmic differentiation their jacobian matrix $J$, and find with the same definition of $t$ as above an equality of $d\times r$
matrices:
$$y_{t+1}\ldots .y_d J=y_{t+1}^{\sum_s\langle a^{t+1},m^s-n^s\rangle}\ldots y_d^{\sum_s
\langle
a^d,m^s-n^s\rangle}(\langle a^j,m^s-n^s\rangle)_{1\leq j\leq d, 1\leq s\leq r}.$$ 
\begin{lemma}\label{regmat} Given an irreducible binomial variety $X\subset {\mathbf A}^d(k)$, with the notations just introduced,  for any regular cone
$\sigma=\langle a^1,\ldots ,a^d\rangle$ compatible with the hyperplanes
$H_\ell$, the rank of the image in
${\rm Mat}_{d\times L}(k)$ of the matrix\break
$(\langle a^j,m^s-n^s\rangle)_{1\leq j\leq d, 1\leq s\leq L}\in {\rm Mat}_{d\times  L}({\mathbf Z})$ is $r$. \end{lemma}
\begin{proof} Since the vectors $a^j$ form a basis of ${\mathbf Q}^d$, and the space $\check {\mathcal W}={\mathcal L}\otimes_{\mathbf Z}{\mathbf R}$  generated
by the $m^s-n^s$ is of dimension $r$, the rank of the matrix
$(\langle a^j,m^s-n^s\rangle)$ is $r$, which proves the lemma if $k$ is of characteristic zero. If the field $k$ is of positive characteristic the proof
is a little less direct; it can be found in [T 2], Chap. 6. Note that in particular the rank of the image in
${\rm Mat}_{d\times r}(k)$  of the matrix $(\langle a^j,m^s-n^s\rangle)_{1\leq j\leq d,1\leq s\leq r}\in {\rm Mat}_{d\times r}({\mathbf Z})$ is $r$.\end{proof}
\begin{lemma}The strict transform $X_1'$ by $\pi (\Sigma)$ of the subspace $X\subset
{\mathbf A}^d(k)$ defined by the ideal of $k[u_1,\ldots ,u_d]$ generated by the binomials $u^{m^1}-\lambda_1 u^{n^1},\ldots
,\break u^{m^r}-\lambda_r u^{n^r}$ is regular and transversal to the exceptional divisor.\end{lemma}
\begin{proof} Let $\sigma$ be a cone of of maximal dimension in the fan $\Sigma$. First note that, in the chart $Z(\sigma)$, none of the coordinates
$y_{t+1},\ldots , y_d$ vanishes on the strict transform $X_1'$ of
$X_1$ and that the equations of $X'_1$ in $Z(\sigma)$ are independent of $y_1,\ldots ,y_t$ . Therefore to prove that the jacobian $J$ of the equations has
rank $r$ at each point of this strict transform it suffices to show that the rank of the image in
${\rm Mat}_{d\times L}(k)$ of the matrix
$(\langle a^j,m^s-n^s\rangle)_{1\leq j\leq d, s\in L}\in {\rm Mat}_{d\times  L}({\mathbf Z})$ is $r$, which follows from the previous
lemma.
\end{proof}
\begin{proposition}\label{binor} The strict transform $X'$ by $\pi (\Sigma)$ of the subspace $X\subset
{\mathbf A}^d(k)$ defined by the ideal of $k[u_1,\ldots ,u_d]$ generated by the $(u^{m^\ell}-\lambda_\ell u^{n^\ell})_{\ell\in \{1,\ldots , L\}}$ is regular and
transversal to the exceptional divisor; it is also irreducible in each chart.
\end{proposition}
\begin{proof} The preceding discussion shows that the rank of $J$ is $r$ everywhere on the
strict transform of $X$, and by Zariski's jacobian criterion this strict transform is smooth and transversal to the exceptional divisor.
 Note however that it is not necessarily irreducible; let us show that the strict transform
of our binomial variety is one of its irreducible components. Since the differences of the exponents in the total transform and the strict transform of a
binomial are the same, the lattice of exponents generated by the exponents of all the strict transforms of the binomials
$(u^{m^\ell}-\lambda_{mn}u^{n^\ell})_{\ell\in \{1,\ldots , L\}}$ is the image
$M(\sigma){\mathcal L}$ of the lattice
${\mathcal L}$ by the linear map
${\mathbf Z}^d\to {\mathbf Z}^d$ corresponding to the matrix
$M(\sigma)$ with rows $(a^1,\ldots ,a^d)$. Similarly the exponents of the strict transforms of $u^{m^1}-\lambda_{m^1n^1}u^{n^1}, \ldots
,u^{m^{r}}-\lambda_{m^rn^r}u^{n^{r}}$ generate the lattice $M(\sigma){\mathcal L}_1$. The lattice $M(\sigma){\mathcal L}$ is the saturation of $M(\sigma){\mathcal
L}_1$, and so according to [E-S], since we assume that $k$ is algebraically closed, the strict transform of our binomial variety is one of the irreducible
components of the binomial variety defined by the $r$ equations displayed above.
\par\noindent
The charts corresponding to regular cones $\sigma\in \Sigma$ of dimension $<d$
are open subsets of those which we have just studied, so they contribute nothing new.\end{proof}
In the case of binomial varieties one can show that the regular refinement $\Sigma$ of the fan $\Sigma_0$ determined by the hyperplanes $H_{m^s-n^s}$ can 
be chosen in such a way that the restriction $X'\to X$ of the map $\pi(\Sigma)$ to the strict transform $X'$ of $X$ induces an isomorphism outside of the
singular locus of $X$; it is therefore an embedded resolution of $X\subset {\mathbf A}^d(k)$ and not only a pseudo-resolution; see [GP-T] and [T2], \S 6.2.\par  
\begin{remark} Since Hironaka's 1964 paper {\rm ([H])}, one usually seeks to achieve resolution of singularities by successions of blowing-ups with non singular centers, which
moreover are ''permissible''. According to {\rm [DC-P]}, toric maps are dominated by finite successions of blowing-ups with non singular centers.
\end{remark}
  Now in view of the results of section
\ref{nondeg}, we expect that if we deform a binomial variety by adding to each of its equations terms which do not affect the Newton polyhedron, the same
toric map will resolve the deformed variety as well. However, it may be only a pseudo-resolution, since the effect of the deformation on the singular locus is
difficult to control. The next section shows that in a special case one can, conversely, present a singularity as a deformation of a toric variety, and thus obtain
an embedded toric resolution.
\section{Resolution of singularities of branches}\label{resb}
This section is essentially  an exposition of material in [G-T] and [T2]. The idea is to show that
any analytically irreducible germ of curve is in a canonical way a deformation of a {\it monomial} curve, which is defined by {\it binomial} equations. In this
terminological  mishap, the {\it monomial} refers to the parametric presentation of the curve; the parametric presentation is more classical, but the binomial
character of the equations is more suitable for resolution of singularities.\par\noindent The deformation from the monomial curve to the curve is ``equisingular'',
so that the toric map which resolves the singularties of the monomial curve according to section \ref{resbin} also resolves the singularities of our original curve once
it is suitably embedded in the affine space where the monomial curve embeds.  One interpretation of this is that {\it after a suitable re-embedding, any analytically
irreducible curve becomes non-degenerate}.
\par For example, in order to resolve the singularities at the origin of the plane curve $C$ with equation
$$(u_1^2-u_0^3)^2-u_0^5u_1=0,$$ a good method is to view it as the fiber for $v=1$ of the family of curves in ${\mathbf A}^3(k)$ defined by the equations
$$\begin{array}{ccl} u_1^2-u_0^3-vu_2&=&0\\
u_2^2-u_0^5u_1&=&0\end{array}$$ as one can see by eliminating $u_2$ between the two equations. The advantage is that the fiber for $v=0$ is a binomial
variety, which we know how to resolve, and its resolution also resolves all the fibers $C_v$. For $v\neq 0$, the fiber $C_v$ is isomorphic to our
original plane curve $C$, re-embedded in ${\mathbf A}^3(k)$ by the functions $u_0,u_1, u_1^2-u_0^3$.\par\noindent In more algebraic terms, it gives
this:\par\medskip  Let
$R$ be a one dimensional excellent equicharacteristic local ring whose completion is an integral domain and whose residue field is algebraically closed. A
basic example is
$R=k[[x,y]]/(f)$ where $k$ is algebraically closed and $f(x,y)$ is irreducible in
$k[[x,y]]$. Then the normalization
$\overline R$ of $R$ is a (discrete) valuation ring because it is a one dimensional normal local ring. The maximal ideal of $\overline R$ is generated by a
single element, say $t$, and each non zero element of
$\overline R$ can be written uniquely as $ut^n$, where $u$ is invertible in $\overline R$ and $n\in {\mathbf N}\cup\{0\}$. The valuation $\nu (ut^n)$ of
that element is $n$.\par\noindent In our basic example, the inclusion
$R\subset \overline R$ is $k[[x,y]]/(f)\subset k[[t]]$ given by $x\mapsto x(t),\ y\mapsto y(t)$ where $x(t),y(t)$ is a parametrization of the plane curve with
equation $f(x,y)=0$. \par\noindent Since
$R$ is a subalgebra of
$\overline R$, the values taken by the valuation on the elements of
$R$ (except $0$) form a semigroup $\Gamma$ contained in ${\mathbf N}$. This semigroup has a finite complement in ${\bf N}$ and is finitely generated.
Let us write it
$$\Gamma=\langle\gamma_0,\gamma_1,\ldots ,\gamma_g\rangle .$$ The powers of the maximal ideal of
$\overline R$ form a filtration 
$$\overline R\supset t\overline R\supset t^2\overline R\supset\cdots\supset t^n\overline R\supset\cdots $$
whose associated graded ring $$\hbox{\rm gr}_\nu \overline R=\bigoplus_{n\in {\bf N}\cup\{0\}} t^n\overline R/t^{n+1}\overline R$$ is a $k$-algebra
isomorphic to the polynomial ring $k[t]$ by the map $t\ (\hbox{\rm mod.}t^2\overline R)\mapsto t$.\par\noindent
This filtration induces a filtration on the ring $R$ itself, by the ideals ${\cal P}_n=R\cap t^n\overline R$, and one defines the corresponding associated graded
ring
$$\hbox{\rm gr}_\nu R=\bigoplus_{n\in {\bf N}\cup\{0\}} {\cal P}_n/{\cal P}_{n+1}\subseteq \hbox{\rm gr}_\nu \overline R=k[t].$$
\begin{proposition} {\rm (see [G-T])} The subalgebra $\hbox{\rm gr}_\nu R$ of $k[t]$ is equal to the subalgebra generated by
$t^{\gamma_0},t^{\gamma_1},\ldots ,t^{\gamma_g}$. It is the semigroup  algebra over $k$ of the semigroup $\Gamma$ of the valuation $\nu$ on $R$; it is
also the affine algebra of the monomial curve in the affine space
${\mathbf A}^{g+1}(k)$ described parametrically by
$u_i=t^{\gamma_i}$ for $0\leq i\leq g$.\end{proposition}
There is a precise geometrical relationship between the original curve $C$ with algebra $R$ and the monomial curve $C^\Gamma$ with algebra $\hbox{\rm
gr}_\nu R$: according to a general principle of algebra, the ring $R$ is a deformation of its associated graded ring. More precisely, let us assume that $R$
contain a field of representatives of its residue field $k$, i.e., that we have a subfield $k\subset R$ such that the composed map $k\subset R\to R/m=k$ is
the identity. This will be the case in particular, according to Cohen's theorem, if the local ring $R$ is complete (and equicharacteristic of course).\par
Start from the filtration by the ideals ${\cal P}_n$ introduced above, set ${\cal P}_n=R$ for $n\leq 0$ and consider the algebra
$${\mathcal A}_\nu (R)=\bigoplus_{n\in {\mathbf Z}}{\cal P}_nv^{-n}\subset R[v,v^{-1}].$$
It can be shown (see [T2]) that it is generated as a $R[v]$-algebra by the $\xi_iv^{-i},\ 0\leq i\leq g$ where $\xi_i\in R$ is of  $t$-adic order $\gamma_i$.
Since ${\mathcal P}_n=R$ for $n\leq 0$ it contains as a graded subalgebra the polynomial algebra $R[v]$, and therefore also $k[v]$. 
\begin{proposition}\label{specia} {\rm (see [T1], [B], Chap. VIII, exercice 2 for \S 6; see also [G1] and [G2])}\par\medskip\noindent a) The composed map
$k[v]\to
{\mathcal A}_\nu (R)$ is faithfully flat.
\par\noindent b) The map $$\sum x_nv^{-n}\mapsto \sum \overline x_n,$$ where $\overline x_n$ is the image of $x_n$  in the quotient ${\cal P}_n/{\cal
P}_{n+1}$, induces an isomorphism $${\mathcal A}_\nu (R)/v{\mathcal A}_\nu (R)\to \hbox{\rm gr}_\nu R.$$\par\noindent
c) For any $v_0\in k^*$ the map $$\sum x_nv^{-n}\mapsto \sum x_nv_0^{-n}$$ induces an isomorphism of $k$-algebras
$${\mathcal A}_\nu (R)/(v-v_0){\mathcal A}_\nu (R)\to R.$$
\end{proposition}
\begin{proof} Since $k[v]$ is a principal ideal domain, to prove a) it suffices (see [B], Chap. I, \S 3, No. 1) to prove that ${\mathcal A}_\nu (R)$ has no torsion as a
$k[v]$-module and that for any
$v_0\in k$ we have $(v-v_0){\mathcal A}_\nu (R)\neq{\mathcal A}_\nu (R)$. The second statement follows from b) and c) which are easy to verify, and the first
follows from the fact that
${\mathcal A}_\nu (R)$ is a subalgebra of $R[v,v^{-1}]$.
\end{proof}
This proposition means that there is a one parameter flat family of algebras whose special fiber is the graded algebra and all other fibers are isomorphic to $R$.
Geometrically, this gives us a flat family of curves whose special fiber is the monomial curve and all other fibers are isomorphic to our given curve. This deformation can be realized in the following way. I assume for simplicity that $R$ is complete. Then by definition of the
semigroup $\Gamma$ there are elements $\xi_0(t),\ldots ,\xi_g(t)$ in $k[[t]]$ which belong to $R$ and are such that their $t$-adic valuations are the generators $\gamma_i$ of the
semigroup $\Gamma$. We may write $\xi_i(t)=t^{\gamma_i}+\sum_{j>\gamma_i}b_{ij}t^j$ with $b_{ij}\in k$. Let us now introduce a parameter $v$ and
consider the family of parametrized curves in ${\mathbf A}^{g+1}(k)$ described as follows:
$$\begin{array}{lr}
u_0&=\xi_0(vt)v^{-\gamma_0}=t^{\gamma_0}+\sum_{j>\gamma_0}b_{0j}v^{j-\gamma_0}t^j\\
u_1&=\xi_1(vt)v^{-\gamma_1}=t^{\gamma_1}+\sum_{j>\gamma_1}b_{1j}v^{j-\gamma_1}t^j\\
\vdots &=\hskip1truecm\vdots \hfill\\
u_g&=\xi_g(vt)v^{-\gamma_g}=t^{\gamma_g}+\sum_{j>\gamma_g}b_{gj}v^{j-\gamma_g}t^j
\end{array}$$
The parametrization shows that for $v=0$ we obtain the monomial curve, and for any $v\neq 0$ a curve isomorphic to our given curve, as embedded in ${\mathbf
A}^{g+1}(k)$ by the functions $\xi_0,\ldots ,\xi_g$. This is a realisation
of the family of Proposition \ref{specia}. In order to get an equational representation of that family, we must begin by finding the equations of the
monomial curve, which we will then proceed to deform.\par
The equations of the monomial curve $C^\Gamma$ correspond to the relations between the generators $\gamma_i$ of $\Gamma$. They are fairly simple in
the case where $\Gamma$ is the semigroup of a plane branch, and in that case $C^\Gamma$ is a complete intersection. 
The general setup is as follows: \par\noindent
Consider the ${\mathbf Z}$-linear map $w\colon {\mathbf Z}^{g+1}\to {\mathbf Z}$ determined by sending the $i$-th base vector $e_i$ to $\gamma_i$; the
image of ${\mathbf Z}_{\geq 0}^{g+1}$ is $\Gamma$. It is not difficult to see that the kernel of $w$ is generated by differences $m-m'$, where $m,m'\in
{\mathbf Z}_{\geq 0}^{g+1}$ and $w(m)=w(m')$. The kernel of
$w$ is a lattice (free sub
${\mathbf Z}$-module)
${\mathcal L}$ in
${\mathbf Z}^{g+1}$, which must be finitely generated because ${\mathbf Z}^{g+1}$ is a n\oe therian ${\mathbf Z}$-module and ${\mathbf Z}$ is a
 principal ideal domain.\par  If we choose a
basis
$m^1-n^1,\ldots ,m^q-n^q$ for
${\mathcal L}$, such that all the $m^j,n^j$ are in ${\mathbf Z}_{\geq
0}^{g+1}$, then it follows from the very construction of semigroup algebras that $C^\Gamma$ is defined in the space
${\mathbf A}^{g+1}(k)$ with coordinates
$u_0,\ldots ,u_g$ by the vanishing of the binomials $u^{m^1}-u^{n^1},\ldots, u^{m^q}-u^{n^q}$.\par Now the faithful flatness of the family of Proposition
\ref{specia} implies that it can be defined in
${\mathbf A}^1(k)\times {\mathbf A}^{g+1}(k)$ by equations which are deformations of the equations of the monomial curve (see [T2], \S 5, proof of 5.49). Here I cheat a
little by leaving out the fact that one in fact defines a formal space. Anyway, our family of (formal) curves is also defined by equations of the form:
$$
\begin{array}{lr}u^{m^1}-u^{n^1}+\sum_{w(r)>w(m^1)}c^{(1)}_r(v)u^r &=0\\
u^{m^2}-u^{n^2}+\sum_{w(r)>w(m^2)}c^{(2)}_r(v)u^r &=0\\ \vdots &\vdots\\  
u^{m^q}-u^{n^q}+\sum_{w(r)>w(m^q)}c^{(1)}_r(v)u^r &=0\end{array}
$$
 where the $c^{(j)}_r(v)$ are in $(v)k[[v]]$,
$w(r)=\sum_0^g\gamma_jr_j$ is the weight of the monomial $u^r$ with respect 
to the weight vector $w=(\gamma_0, \ldots ,\gamma_g)$, i.e., $w(r)=\langle w,r\rangle$. Remember that by construction $w(m^i)=w(n^i)$ for $1\leq i\leq q$.
This means that we deform each binomial equations by adding terms of  weight greater than that of the binomial . It is shown in [T2] that the parametric representation and the
equational representation both describe the deformation of proposition \ref{specia}. Up to completion with respect to the $(u_0,\ldots ,u_g)$-adic topology, the algebra ${\mathcal
A}_\nu(R)$ is the quotient of
$k[v][[u_0,\ldots ,u_g]]$ by the ideal generated by the equations written above. It is also equal to the subalgebra $k[[\xi_0(vt)v^{-\gamma_0},\ldots
,\xi_g(vt)v^{-\gamma_g}]]$ of $k[v][[t]]$.\par\noindent  One may remark that, in the case where the $\xi_j(t)$ are polynomials, there is a close analogy with
the SAGBI algebras bases for the subalgebra
$k[\xi_0(t),\xi_1(t)]\subset k[t]$ (see [St]). This is developed in the text [Br] of Anna Bravo. 
\par\noindent This equational description is the generalization of the example shown at the beginning of this section.\par\noindent Now it should be more
or less a computational exercise to check that a toric map $Z(\Sigma)\to {\mathbf A}^{g+1}$ which resolves the binomial variety $C^\Gamma$ also resolves the
``nearby fibers'', which are all isomorphic to $C$ re-embedded in ${\mathbf A}^{g+1}$. There is however a difficulty (see [G-T]) which requires the use of Zariski's
main theorem.\par The results of this section have been extended by P. Gonz\'alez P\'erez in [GP] to the much wider class of irreducible \textit{quasi-ordinary} germs of hypersurface singularities,
whose singularities are not isolated in general.\par This shows that a toric resolution of binomial varieties can be used, by considering suitable deformations, to resolve singularities which
are at first sight far from binomial.
\section{\textbf Appendix:}\label{mult}\par\medskip\noindent
\centerline{\bf  MULTIPLICITIES, VOLUMES, AND NON DEGENERACY}
\subsection{Multiplicities and volumes}
\par\medskip 
One of the interesting features of the Brian\c con-Skoda theorem is that it provides a way to pass from the integral
closure of an ideal to the ideal itself, while it is much easier to check that a given element is in the integral closure of an ideal than to
check that it is in the ideal. For this reason, the theorem has important applications in problems of effective commutative algebra
motivated by transcendental number theory. In the same vein, this section deals, in the monomial case, with the problem of
determining from numerical invariants whether two ideals have the same integral closure, which is much easier than to determine
whether they are equal. The basic fact coming to light is that multiplicities in commutative algebra are like volumes in the theory of
convex bodies, and indeed, for monomial ideals, they \textit{are} volumes, up to a $d!$ factor (compare with [T3]). The same is true for degrees of invertible sheaves on algebraic
varieties. Exactly as monomial ideals, and for the same reason, the degrees of equivariant invertible sheaves generated by their global sections on toric
varieties \textit{are} volumes of compact convex bodies multiplied by $d!$ (see [T1]). In this appendix proofs are essentially replaced by references.

Let $R$ be a n\oe therian ring, and ${\mathbf q}$ an ideal of $R$ such that the $R$-module $R/{\mathbf q}$ has finite length $\ell_R (R/{\mathbf
q})=\ell_{R/{\mathbf q}}(R/{\mathbf q})$. Then the quotients ${\mathbf q}^n/{\mathbf q}^{n+1}$ have finite length as $R/{\mathbf q}$-modules and one can
define the Hilbert-Samuel series (for all this, see [B], Chap. VIII, \S 4)
$$H_{R,\mathbf q}=\sum_{n=0}^\infty\ell_{R/{\mathbf q}} ({\mathbf q}^n/{\mathbf q}^{n+1})T^n\in {\mathbf Z}[[T]].$$
According to \textit{loc.cit.}, there exist an integer $d\geq 0$ and an element $P\in {\mathbf Z}[T,T^{-1}]$ such that $P(1)>0$ and $$H_{R,\mathbf q}=(1-T)^{-d}P.$$
From this follows the:
\begin{proposition}{\rm (Samuel)} Given $R$ and ${\mathbf q}$ as above, there exist an integer $N_0$ and a polynomial $Q(U)$ with rational coefficients
such that for
$n\geq N_0$ we have
$$\ell_{R/{\mathbf q}} (R/{\mathbf q}^n)=Q(n).$$\end{proposition}
If we assume that ${\mathbf q}$ is primary for some maximal ideal $m$ of $R$, i.e., ${\mathbf q}\supset m^k$ for large enough $k$, the degree of the
polynomial
$Q$ is the dimension
$d$ of the local ring
$R_m$, and the highest degree term of $Q(U)$ can be written $e({\mathbf q},R)\frac{U^d}{d!}$. In fact (see \textit{loc.cit.}), we have
$e({\mathbf q},R)=P(1)\in {\mathbf N}$.\par By definition, the integer $e({\mathbf q},R)$ is the {\it multiplicity} of the ideal ${\mathbf q}$ in $R$.\par\noindent
If $R$ contains a field $k$ such that $k=R/m$, we can replace $\ell_{R/{\mathbf q}}(R/{\mathbf q}^n)$ by its dimension $\hbox{\rm dim}_k(R/{\mathbf q}^n)$ as a $k$-vector
space.\par Let us take $R=k[u_1,\ldots ,u_d]$ and ${\mathbf q}=(u^{m^1},\ldots ,u^{m^q})R$; the ideal ${\mathbf q}$ is primary for the maximal ideal $m=(u_1,\ldots
,u_d)R$ if and only if $\hbox{\rm dim}_kR/{\mathbf q}<\infty$. Now one sees that the images of the monomials $u^m$ such that $m$ is not contained in
$E=\bigcup_{i=1}^q(m^i+{\mathbf R}_{\geq 0}^d)$ constitute a basis of the $k$-vector space $R/{\mathbf q}$:
$$\hbox{\rm dim}_kR/{\mathbf q}=\#{\mathbf Z}^d\cap ({\mathbf R}_{\geq 0}^d\setminus E) .$$ For the same reason we have for all $n\geq 1$, since ${\mathbf q^n}$ is also
monomial,
$$\hbox{\rm dim}_kR/{\mathbf q^n}=\#{\mathbf Z}^d\cap ({\mathbf R}_{\geq 0}^d\setminus nE) ,$$ where $nE$ is the set of sums of $n$ elements of $E$.\par
From this follows, in view of the polynomial character of the first term of the equality, the:
\begin{corollary} Given a subset $E=\bigcup_{s=1}^q(m^s+{\mathbf R}_{\geq 0}^d)$ whose complement in  ${\mathbf R}_{\geq 0}^d$ has finite volume, there exists an integer
$N_0$ and a polynomial $Q(n)$ of degree $d$ with rational coefficients such that for $n\geq N_0$ we have
$$\#{\mathbf Z}^d\cap ({\mathbf R}_{\geq 0}^d\setminus nE)=Q(n).$$\end{corollary}
Therefore, we have:
$$\lim_{n\to\infty}\frac{Q(n)}{n^d}=\lim_{n\to\infty}\frac{\#{\mathbf Z}^d\cap ({\mathbf R}_{\geq 0}^d\setminus
nE)}{n^d}=\lim_{n\to\infty}\hbox{\rm Covol}\frac{nE}{n}=\hbox{\rm Covol}{\overline E},$$ where $\hbox{\rm Covol}A$, the \textit{covolume} of $A$, is the volume of the
complement of $A$ in ${\mathbf R}_{\geq 0}^d$. The last equality follows from the remark made in section \ref{BrSk}, and the previous one from the classical fact of calculus that as
$n\to\infty$, $\hbox{\rm Covol}\frac{nE}{n}=\frac{\#{\mathbf Z}^d\cap ({\mathbf R}_{\geq 0}^d\setminus nE)}{n^d}+o(1)$. Since the limit as
$n\to\infty$ of $\frac{Q(n)}{n^d}$ is $\frac{e({\mathbf q},R)}{d!}$, we have immediately the:
\begin{corollary}\label{convmul} For a monomial ideal ${\mathbf q}=(u^{m^1},\ldots ,u^{m^s})$ in $R=k[u_1,\ldots, u_d]$ which is primary for $m=(u_1,\ldots, u_d)$ , with the
notations above, we have the two equalities$$\begin{array}{lrl}\hbox{\rm dim}_k(R/{\mathbf q})&=&\#{\mathbf Z}^d\cap ({\mathbf R}_{\geq 0}^d\setminus E)\cr &&\cr
e({\mathbf q},R)&=&d!\hbox{\rm Covol}\overline E.\cr\end{array}$$
\end{corollary}
\begin{corollary}\label{monomrees}{\rm (Monomial Rees Theorem, an avatar of [R1])}\par\noindent a) For a monomial primary ideal ${\mathbf q}$ as above, me have
$$e(\overline{\mathbf q},R)=e({\mathbf q},R).$$
b) Given two such ideals ${\mathbf q}_1,{\mathbf q}_2$ such that ${\mathbf q}_1\subseteq{\mathbf q}_2$, we have $\overline{\mathbf
q}_1=\overline{\mathbf q}_2$ if and only if $e({\mathbf q}_1,R)=e({\mathbf q}_2,R)$.\end{corollary}
These results hold (Rees' theorem, see [R1]) for ideals containing a power of the maximal ideal in a n\oe therian local ring $R$ whose completion is equidimensional. \par
Now there is a well know theorem of Minkowski in the theory of convex bodies, concerning the volume of the Minkowski sum of compact convex sets.
Recall that for $K_1,K_2$ in ${\mathbf R}^d$, the \textit{Minkowski sum} $K_1+K_2$ is the set of sums $\{x_1+x_2; \ x_1\in K_1,x_2\in K_2\}$ and for $\lambda\in
{\mathbf R}$,
$\lambda K=\{\lambda x;\ x\in K\}$ :
\begin{theorem}{\rm(Minkowski)} Given $s$ compact convex subset $K_1,\ldots ,K_s$ of  ${\mathbf R}^d$, there is a homogeneous expression for the
$d$-dimensional volume of the positive Minkowski linear combination of the $K_i$, with $(\lambda_i)_{1\leq i\leq d}\in {\mathbf R}^d_{\geq
0}$: 
$$\hbox{\rm Vol}_d(\lambda_1K_1+\cdots +\lambda_sK_s)= \sum_{\sum_1^s\alpha_i=d}\frac{d!}{\alpha_1!\ldots\alpha_s!}\hbox{\rm
Vol}(K_1^{[\alpha_1]},\ldots ,K_s^{[\alpha_s]})\lambda_1^{\alpha_1}\ldots\lambda_s^{\alpha_s},$$ where the coefficients $\hbox{\rm
Vol}(K_1^{[\alpha_1]},\ldots ,K_s^{[\alpha_s]})$ are non negative and are called the {\it mixed volumes} of the convex sets $K_i$.\par\noindent Taking $s=2$
gives
$$\hbox{\rm Vol}_d(\lambda_1K_1+\lambda_2K_2)=\sum_{i=0}^d{d\choose i}\hbox{\rm Vol}(K_1^{[i]}
,K_2^{[d-i]})\lambda_1^i\lambda_2^{d-i}.$$\end{theorem} The proof is obtained by approximating the convex bodies by polytopes, and using the Cauchy
formula for the volume of polytopes. Exactly the same proof applies to the covolumes of {\it convex} subsets of ${\mathbf R}_{\geq 0}^d$ to give the corresponding
theorem:
$$\hbox{\rm Covol}_d(\lambda_1\overline E_1+\cdots +\lambda_s\overline E_s)= \sum_{\sum_1^s\alpha_i=d}\frac{d!}{\alpha_1!\ldots\alpha_s!}\hbox{\rm
Covol}(\overline E_1^{[\alpha_1]},\ldots ,\overline E_s^{[\alpha_s]})\lambda_1^{\alpha_1}\ldots\lambda_s^{\alpha_s},$$ defining the \textit{mixed
covolumes} of such convex subsets.\par On the other hand, there is an analogous formula  in commutative algebra:
\begin{theorem}{\rm (see [T6]):} Given ideals ${\mathbf q}_1,\ldots ,{\mathbf q}_s$ which are primary for a maximal ideal $m$ in a n\oe therian ring $R$
such that the localization $R_m$ is a
$d$-dimensional local ring and the residue field $R_m/mR_m$ is infinite, there is for $\lambda_1,\ldots ,\lambda_s\in {\mathbf Z}_{\geq 0}^s$ an expression 
$$e({\mathbf q}_1^{\lambda_1}\ldots {\mathbf
q}_s^{\lambda_s},R)=\sum_{\sum_1^s\alpha_i=d}\frac{d!}{\alpha_1!\ldots\alpha_s!}e({\mathbf q}_1^{[\alpha_1]},\ldots
,{\mathbf q}_s^{[\alpha_s]};R)\lambda_1^{\alpha_1}\ldots\lambda_s^{\alpha_s},$$ where the coefficients $e({\mathbf q}_1^{[\alpha_1]},\ldots
,{\mathbf q}_s^{[\alpha_s]};R)$ are non negative integers and are called the {\it mixed multiplicities} of the primary ideals ${\mathbf q}_i$. This name is
justified by the fact that $e({\mathbf q}_1^{[\alpha_1]},\ldots ,{\mathbf q}_s^{[\alpha_s]};R)$ is the multiplicity of an ideal generated by $\alpha_1$ elements
of ${\mathbf q}_1$,$\dots$, $\alpha_s$ elements
of ${\mathbf q}_s$, chosen in a sufficiently general way.
 Taking $s=2$ gives
$$e({\mathbf q}_1^{\lambda_1} {\mathbf
q}_2^{\lambda_2},R)=\sum_{i=0}^d{d\choose i}e({\mathbf q}_1^{[i]},{\mathbf q}_2^{[d-i]};R)\lambda_1^i\lambda_2^{d-i}.$$\end{theorem}
from this and Corollary \ref{convmul} follows immediately:
\begin{corollary}\label{mixequal} Let $k$ be an infinite field. Given monomial ideals ${\mathbf q}_1,\ldots ,{\mathbf q}_s$ which are primary for the
maximal ideal
$(u_1,\ldots, u_d)$ in
$R=k[u_1,\ldots, u_d]$, and denoting by $E_i$ the corresponding subsets generated by their exponents, we have for all $\alpha\in {\mathbf Z}_{\geq 0}^d$ such
$\sum_1^s\alpha_i=d$ the equality 
$$e({\mathbf q}_1^{[\alpha_1]},\ldots
,{\mathbf q}_s^{[\alpha_s]};R)=d!\hbox{\rm
Covol}(\overline E_1^{[\alpha_1]},\ldots ,\overline E_s^{[\alpha_s]}).$$\end{corollary}
In particular, the mixed multiplicities depend only on the integral closures of the ideals ${\mathbf q}_i$.
Now there are well known inequalities for the mixed volumes of two compact convex bodies: the Alexandrov-Fenchel inequalities:
\begin{theorem}\label{AlexF}  {\rm (Alexandrov-Fenchel; see [Gr])} a)  Let $K_1,K_2$ be two compact convex bodies in ${\mathbf R}^s$; set
$v_i=\hbox{\rm Vol}(K_1^{[i]},K_2^{[d-i]})$. For all
$2\leq i\leq d$ the following inequalities hold:
$$v_{i-1}^2\geq v_iv_{i-2}.$$
b) Equality holds in all these inequalities if and only if  for some $\rho\in {\mathbf R}_+$ we have $K_1=\rho
K_2$ up to translation. If all the $v_i$ are equal, then $K_1=K_2$ up to translation, and conversely.
\end{theorem}
Let ${\mathbf B}^d$ denote the
$d$-dimensional unit ball, and $A$ any subset of ${\mathbf R}^d$ which is tame enough for the volumes to exist.\par\noindent
The problem which inspired this theorem is to prove that in the isoperimetric inequality $$\hbox{\rm
Vol}_{d-1}(\partial A)^d\geq d^d\hbox{\rm Vol}_d({\mathbf B}^d)\hbox{\rm Vol}_d(A)^{d-1},$$ equality should hold only if $A$ is a multiple of the unit ball, to
which some  ''hairs'' of a smaller dimension than $\partial A$ have been added. In the case where $A$ is convex, taking $K_1$ to be the unit ball and $K_2=A$, one
notices that
$v_0=\hbox{\rm Vol}_d(A)$ and $v_1=d^{-1}\hbox{\rm Vol}_{d-1}(\partial A)$; the isoperimetric inequality then follows very quickly by an appropriate
telescoping of the Alexandrov-Fenchel inequalities. From this telescoping follows the fact that if we have equality in the isoperimetric inequality for a convex
subset $A$ of ${\mathbf R}^d$, then we have equality in \textit{all} the Alexandrov-Fenchel inequalities for $A$ and the unit ball, so that $A$ must be a ball. By the same type
of telescoping, one proves the inequalities
$v_i^d\geq v_0^{d-i}v_d^i$, which gives the:
\begin{theorem}\label{BrMin} {\rm (Br\"unn-Minkowski; see [Gr])} For convex compact subsets of ${\mathbf R}^d$, the following inequality holds
$$\hbox{\rm Vol}_d(K_1+K_2)^{\frac{1}{d}}\geq \hbox{\rm Vol}_d(K_1)^{\frac{1}{d}}+\hbox{\rm Vol}_d(K_2)^{\frac{1}{d}},$$
with equality if and only if $K_1$ and $K_2$ are homothetic up to translation, or $\hbox{\rm Vol}_d(K_1+K_2)=0$, or one of them is a point.\end{theorem} 
The same constructions and proof apply to covolumes, where the inequalities are reversed; they correspond to inequalities for the mixed multiplicities of
monomial ideals, which are in fact true for primary ideals in formally equidimensional n\oe therian local rings:
\begin{theorem}\label{mixed} {\rm (See [T4], [R-S], [T5], [K])} a)  Let ${\mathbf q}_1,{\mathbf q}_2$ be two primary ideals in the n\oe therian local ring
$R$ ; set
$w_i=e({\mathbf q}_1^{[i]},{\mathbf q}_2^{[d-i]};R)$. For all
$2\leq i\leq d$ the following inequalities hold:
$$w_{i-1}^2\leq w_iw_{i-2}.$$
b) Assuming in addition that $R$ is formally equidimensional (quasi-unmixed), equality holds in all these inequalities if and only if  for some $a,b\in
{\mathbf N}$ we have
$\overline{{\mathbf q}_1^a}=\overline{{\mathbf q}_2^b}$. If all the $w_i$ are equal, then $\overline{{\mathbf q}_1}=\overline{{\mathbf q}_2}$, and
conversely.\end{theorem} So in this case again, the combinatorial inequalities appear as the avatar for monomial ideals of general inequalities of
commutative algebra. One can see that if ${\mathbf q}_1\subseteq {\mathbf q}_2$, we have $e({\mathbf q}_1, R)=w_d \geq w_i\geq w_0=e({\mathbf
q}_2,R)$, for $1\leq i\leq d-1$. So this result implies Rees' Theorem, which is stated after Corollary \ref{monomrees}.\par In fact the same happens for the Alexandrov-Fenchel
inequalities, which are the avatars for toric varieties associated to polytopes of general inequalities of K\"ahler geometry known as the Hodge Index Theorem. This is due to the fact that
the mixed volumes of rational convex polytopes are equal, up to a
$d!$ factor, to the mixed degrees of invertible sheaves (or of divisors) on certain toric varieties associated to the collection of polytopes, exactly as in Corollary
\ref{mixequal}. This approach to Alexandrov-Fenchel inequalities was introduced by Khovanskii and the author; see [Gr] for an excellent exposition of this topic, and [Kh], 
[T1].\par In all these cases, a remarkable fact is that, thanks to the positivity and convexity properties of volumes (resp., multiplicities), a
\textit{finite} number of equations on a pair $(A_1,A_2)$ of objects in an
\textit{infinite dimensional} space (convex bodies modulo translation or integrally closed primary ideals) suffices to ensure that $A_1=A_2$.
\subsection{Newton non degenerate ideals in $k[[u_1,\ldots ,u_d]]$ and\newline multiplicities}
Define the support $S(I)$ of an ideal $I$ of $k[[u_1,\ldots ,u_d]]$ to be the set of the exponents $m$ appearing as one of the exponents in at least one series
belonging to the ideal
$I$. Define the Newton polyhedron ${\cal N}_+(I)$ of $I$ as the boundary of the convex hull ${\cal P}_+(I)$ of $\bigcup_{m\in S(I)}(m+{\mathbf R}_{\geq
0}^d)$.\par\noindent According to [B-F-S], a primary ideal
${\mathbf q}$ is said to be \textit{non degenerate} if it admits a system of generators $q_1,\ldots ,q_t$ such that their restrictions to each compact face of ${\cal
N}_+(I)$ have no common zero in the torus $(k^*)^d$. The following is part of what is proved in [B-F-S], \S 3:
\begin{theorem}{\rm ([B-F-S])} For a primary ideal ${\mathbf q}$ of $R=k[[u_1,\ldots ,u_d]]$, the following conditions are equivalent:\par\noindent
a) The ideal ${\mathbf q}$ is non degenerate in the coordinates $u_1,\ldots ,u_d$,\par\noindent
b) The equality $e({\mathbf q},R)=d!\hbox{\rm Covol}{\cal P}_+(I)$ holds,\par\noindent
c) The integral closure $\overline{\mathbf q}$ of ${\mathbf q}$ is generated by monomials in $u_1,\ldots ,u_d$.
\end{theorem}
It follows from this that monomial ideals are non degenerate, and that products of non degenerate primary ideals are non degenerate ([B-F-S], Corollary
3.14). Moreover, all the numerical facts mentioned above for monomial ideals with respect to their Newton polyhedron are valid for non degenerate ideals
(\textit{loc.cit.}). Non degenerate ideals behave as \textit{reductions}  of monomial ideals, which in fact they are. Here we can think of a reduction (in the sense of
Northcott-Rees; see [N-R], [R3]) of an ideal ${\cal M}\subset  k[[u_1,\ldots ,u_d]]$ as an ideal generated by $d$ sufficiently general combinations of generators
of ${\cal M}$. More precisely, it is an ideal ${\cal M}'$ contained in ${\cal M}$ and having the same integral closure.
\par\noindent There is a close connection between this non degeneracy for ideals and the results of section
\ref{nondeg}; if the ideal
${\mathbf q}=(q_1,\ldots ,q_s)k[[u_1,\ldots ,u_d]]$ is non degenerate, then a general linear combination $q=\sum_{i=1}^s\lambda_iq_i$ is non degenerate with
respect to its Newton polyhedron.
\par\medskip There are
many other interesting consequences of the relationship between monomial ideals and combinatorics; I refer the reader to [St]. \par\noindent
All the results of this appendix remain valid if $k[u_1,\ldots ,u_d]$ and its completion $k[[u_1,\ldots ,u_d]]$ are replaced respectively by
$k[\check\sigma\cap{\mathbf Z}^d]$ and its completion, for a strictly convex cone $\sigma\subset {\mathbf R}_{\geq 0}^d$.\par There are also generalizations of mixed
multiplicities to collections of not necessarily primary ideals (see [R2]) and to the case where one of the ideals is replaced by  a submodule of finite colength of a free
$R$-module of finite type (see [K-T]).\par\noindent
\par\noindent It would be interesting to determine how the results of this appendix extend
to monomial submodules of a free
$k[u_1,\ldots ,u_d]$-module.
%--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

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\end{thebibliography}

%-----------------------------------------------
\bigskip\noindent
Institut math\'ematique de Jussieu,\newline 
UMR 7586 du C.N.R.S.,\newline
%Universit\'es Paris 6-Pierre et Marie Curie et Paris 7-Denis Diderot,\newline 
%Case 7012, 2 Place Jussieu,\newline
%75251 Paris Cedex 05, France\newline
16, Rue Clisson, 75013 Paris, France\newline
teissier@math.jussieu.fr
\end{document}

