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\centerline{\bf Errata and addenda for ``Valuations, deformations, and toric
geometry''}
\centerline{Bernard Teissier}\par\medskip\noindent
page 362, table of contents, section 4, read: ...valuation rings and valued n\oe therian rings.\par\noindent
page 366, line -15, read: ... by Abhyankar's inequality we have $\hbox{\rm r}(\nu)\leq\hbox{\rm dim}R$.\par\noindent
page 369, footnote, the last letters in both words of the first Greek
quotation are sigmas. \par\noindent
page 371, lines 11-13: it is implicitely assumed that each $\overline
x_\phi$ is $\neq 0$.
\par\noindent 
page 374, line 3, the correct equality is 
$$\A_\nu(R)/(\bigoplus_{\phi \in \Phi_+}\P_\phi (R)v^{-\phi}+m\A_\nu (R))\simeq k[v^{\Phi_+}]$$
\par\noindent 
page 384, line -20, read: ``...which can also be deduced from subsection 3.3''
\par\noindent 
page 389, title of section 4, read: ...valuation rings and valued n\oe therian rings.
\par\noindent 
page 390, line 14, read: ...with $c_{m,m'}\in k,\ \ \lambda_{m,m'}\in k^*$.\par\noindent 
page 392, line -14, read: ``...version of (non embedded) local uniformization;''
\par\noindent
$\bullet$ page 393, line 1, after the reference to {Z1}, add: see also [H-P], Chap. XVIII, \S 5.
\par\noindent
page 397, line -3, read:..irrational and $>1$,
\par\noindent
page 398, lines 20,21, read: where $U_1=V_2$ is of degree $1$ and $U_2=V_1$ of degree $\tau$.\par\noindent Addendum:  In the notations of the Perron algorithm, the degree of $V_i$ is $\tau_2^{(i)}$ and the degree of $V_{i+1}$ is $\tau_1^{(i)}$.
\par\noindent
page 401, line 18, read: $P(u_2,u_3)$. \par\noindent
page 401, footnote: the third letter of the Greek word is an upsilon
and the last letter  is a sigma.
\par\noindent 
page 415, lines 4 and 5 of the proof of Lemma 5.25, replace $(X_i)_{i\in I}$ by $(U_i)_{i\in I}$, $cX^a$ by $cU^a$, $X^a$ by $U^a$.
\par\noindent 
page 415, in Example 5.27, read:
$$x-\widetilde{\hbox{\rm in}}_\nu x=(a-\tilde {a})f^\ell \ \ \hbox{\rm mod.}f^{\ell +1}$$
\par\noindent 
page 415, last line of the proof of lemma 5.25: replace ``the sequence $x^{(j)}$'' by `` the sequence $x-x^{(j)}$ of elements of $k[(\xi_i)_{i\in I}]$''.\par\noindent
page 416, line -6, add a $(-1)^n$ before the last sum sign in the formula.
\par\noindent
page 417
From ``given in advance'', and until ``we may now repeat'',  replace the text by:

Let $s$ be the least integer such that for
infinitely many values of the integer $j$ we have $\nu_s(x-x^{(j+1)})>\nu_s(x-x^{(j)})$. If $s=1$,
since the valuation $\nu_1$ is of height one, for any $\phi_1\in \Phi_{1+}$, after finitely
many steps we have $\nu_1 (x-x^{(j)})>\phi_1$, hence for any $\phi_1\in \Phi_{1+}$, after finitely
many steps we have $\nu(x-x^{(j)})>\phi_1$, so we may take $r_j=x-x^{(j)}$ and get $\nu
(x-\sum_{r=0}^{j}P_r(\xi_i))= \nu (x-r_j)>\phi$ and this shows that $x\in \hat R_h$. Assume now $s>1$, and let $x^{(1)}$ be the representative in $\hat R_{s-1}[(\xi_i)_{i\in I_s}]$ of the initial form $\hbox{\rm in}_{\nu_{s-1}}(x)\in \hbox{\rm gr}_{\nu_{s-1}} R $. This makes sense because we may assume by induction on the dimension of $R$ that the subring $\hat R_{s-1}\subset R$ is a representative of  $R/(m_{\nu_{s-1}}\cap R)$. We define inductively the $x^{(j)}$ as representatives in $\hat R_{s-1}[(\xi_i)_{i\in I_s}]$ of $\hbox{\rm in}_{\nu_{s-1}}(x-x^{(j-1)})$. By
definition of $s$, there exists a $j_0$ such that $\nu_{s-1}(x-x^{(j+1)})= \nu_{s-1}(x-x^{(j)})$ for $j\geq
j_0$. By substracting from $x$ a polynomial in the $\xi_i$, we may assume that
$j_0=1$.\par\noindent Note that the center of $\nu_{s-1}$ is necessarily distinct from the
center of $\nu_s$, since if the two centers are equal, there are only finitely many distinct
$\nu_s$-ideals between two consecutive $\nu_{s-1}$-ideals of $R$, according to [Z-S], Vol.
II, Appendix 3, Corollary p.345. \par\noindent We consider the initial forms
$\hbox{\rm in}_{\nu_{s-1}}(x- x^{(j)})\in \hbox{\rm gr}_{\nu_{s-1}} R $; for $j\geq j_0$; they all
have the same degree, say $\phi_{s-1}$ and they are in $\P_{\phi_{s(j)}}/\P^+_{\phi_{s-1}}$ with $\phi_s(j)$ increasing strictly with $j$. The $R/{\bf p}_{s-1}$-submodules $\P_{\phi_s}/\P^+_{\phi_{s-1}}\subset \P_{\phi_{s-1}}/\P^+_{\phi_{s-1}}$ form a simple infinite sequence in view of Proposition 3.17. This sequence of submodules has intersection $(0)$ and by the module-theoretic version of Chevalley's theorem ([B3], Chap. IV, \S 2, No.5, Cor.4) there is a sequence of integers $t(\phi_s(j))$ tending to infinity with the image of $\phi_s(j)$ in the height one group $\Psi_{s-1}/\Psi_s$ and such that $\P_{\phi_s(j)}/\P^+_{\phi_{s-1}}\subset m^{t(\phi_s(j))}(\P_{\phi_{s-1}}/\P^+_{\phi_{s-1}})$. Since each
homogeneous component $\P_{\phi_{s-1}}/\P^+_{\phi_{s-1}}$ of $\hbox{\rm gr}_{\nu_{ s-1}} R $ is a $R/(m_{\nu_{s-1}}\cap
R)$-module of finite type, it is complete for the $m/(m_{\nu_{s-1}}\cap R)$-topology.
It is also complete for the $\overline\nu_s$ topology by (the proof of) Proposition 5.10. The sequence of the initial forms $\hbox{\rm in}_{\nu_{s-1}}(x-x^{(j)})$ then converges in $(\hbox{\rm gr}_{\nu_{s-1}}
R)_{\phi_{s-1}}$ for the $m$-adic topology, and therefore also for the topology defined by the $\P_{\phi_s}/\P^+_{\phi_{s-1}}$,  to a unique limit $\overline {x_1^{(1)}}$. By the definition of $\hat R_s$, we can lift  $\overline {x_1^{(1)}}$ to an element
$x_1^{(1)}\in
\hat R_s$. This element has the property that for all $j\geq 1$, we have $\nu_{s-1}(x-x_1^{(1)})>\nu_{s-1}(x-x^{(j)})=\nu_{s-1}(x)$. Replacing now $x$ by $x-x_1^{(1)}$, and repeating this construction, we build a sequence of elements
$x_1^{(q)}\in \hat R_h$ such that $\nu_{s-1} (x-x_1^{(q)})$ increases at each step. 
\par\noindent \
page 418, from the beginning of 5.3 to: ``This is a $\nu$-adic..'', replace the text by:\par\noindent
Given a system $(\overline \xi_i)_{i\in I}$ of generators of the $k$-algebra
$\hbox{\rm gr}_\nu R$, giving rise to a surjective map $k[(U_i)_{i\in I}]\to \hbox{\rm gr}_\nu R$, a field of representatives $k\subset \hat
R^{(\nu)}$ and representatives
$(\xi_i)_{i\in I}$ in $R$ of the $\overline\xi_i$, the main result of this subsection is the extension of the map $k[(u_i)_{i\in I}]\to \hat R^{(\nu)}$ mapping $u_i$ to $\xi_i$ to a continuous
surjective map
$$\widehat{k[(w_j)_{j\in J}]}\to \hat R^{(\nu)}$$ from the scalewise completion of the polynomial ring to the scalewise $\nu$-adic
completion of $R$. When $\widehat{k[(w_j)_{j\in J}]}$ is endowed with the term order obtained by giving to $w_j$ the valuation of its image
$\eta_j\in \hat R^{(\nu)}$, the associated graded map is the map 
$$k[(W_j)_{j\in J}]\to \hbox{\rm gr}_{\hat\nu} \hat R^{(\nu)}$$
mapping $W_j$ to $ \overline\eta_j$, where each $ \overline\eta_j$ is a Laurent monomial in the $\overline\xi_i$. 
\par\noindent
page 424, replace line 5 by: Using the fact that $\widehat{\A_\nu (R)}^{(\nu_\A)}$ is a faithfully flat $k[v^{\Phi_+}]$-algebra, one can show:
\par\noindent 
page 425, replace line 6 of `Corollary 5.51 by:
$$\hat c\colon \widehat{k[(w_j)_{j\in J}]}
\to \hat R^{(\nu )} ,\ \ \ w_j\mapsto c(-\hat\nu(\eta_j))\eta_j$$\par\noindent
page 426, first line of the proof of Cor. 5.53: delete ''of
the''.\par\noindent
page 426, change the statement of Cor. 5.54 to:\par\noindent
{\it Let $R$ be a complete noetherian local ring endowed with a rational valuation $\nu$ of height $\hbox{\rm h}(\nu)$. In the minimal system of
generators of the semigroup $\Gamma$ of $(R,\nu)$ there are at most $\hbox{\rm h}_R(\nu)-1$ elements without predecessor.}
\par\noindent
page 426, first line of the proof of Cor. 5.54, read:..height $\hbox{\rm h}_R(\nu)$ of $\nu$ in $R$.
\par\noindent
page 426, at the end of the proof of Cor. 5.54, read ..height of $\nu$ in $R$ minus one.\par\noindent
page 427 line -4, replace $\partial_{u_{h+1}}$ by $\partial_{w_h}$.
\par\noindent
page 444 line 13, read : $\hbox{\rm Spec}k[v][[u_1,\ldots ,u_N]]$\par\noindent
page 451, first line of the statement of Corollary 7.7, read : $\hbox{\rm gr}_{\hat\nu} \hat R^{(\nu)}$ and not $\hbox{\rm gr}_\nu\hat R^{(\nu)}$.
\par\noindent 
$\bullet$ page 457, after [H2], add \par\noindent 
[H-P]  W.V.D. Hodge and D. Pedoe, {\it Methods of Algebraic Geometry, Vol. 3), Cambridge mathematical library, Cambridge University Press.
\par\noindent 
page 457,
reference [Kr], the year is 1932.
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