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\begin{document}
\bibliographystyle{amsplain}
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\title{Erratum to "Monomial ideals, binomial ideals, polynomial ideals"}
\author{Bernard Teissier}
\address{Institut de Math\'ematiques de Jussieu - Paris Rive Gauche, UMR 7586 du CNRS, 
B\^atiment Sophie Germain, Case 7012,
75205 PARIS Cedex 13, France}

\email{bernard.teissier@imj-prg.fr}

\keywords{Toric geometry, integral closure. binomial ideals}


\pagestyle{myheadings}
%\markboth{\rm Bernard Teissier}{\rm Overweight}


\subjclass[2000]{14M25, 14E15, 14B05}


\dedicatory{ \textit {}}
\maketitle
\begin{abstract} Patrick Popescu Pampu has called my attention to several errors in the text \cite{Te}. I am grateful to him. Page and line numbers refer to the version on this site, those between parenthesis to the  published version. \end{abstract}
\begin{itemize}
\item p. 6, line 6 (p. 216, line 9) (Corrected in the on-site version): $\dots$together with their faces. 
\item p. 14, line 8 (p. 223, line -18) (Corrected in the on-site version):$\dots$parallel to coordinate planes (intersections of coordinate hyperplanes).
\item p. 14, line -7 (p. 224, line 7)(Corrected in the on-site version)
: $\dots$dual fan (except for the origin) of a Newton$\dots$
\item p.17 (p. 227) (Corrected in the on-site version, where the exercise is replaced by a remark.) : the exercise is wrong; one does not obtain, by varying $\check a\in \R^d_{>0}$ all collections of compact faces of the polyhedra of $f_1,\ldots ,f_k$. Here is Popescu-Pampu's example, with $d=3,k=2$:
$$f_1=x^2+y^2+z^3, \ f_2=x^2+\eta y^3+\zeta z^4\ \ {\rm with}\ \eta,\zeta\in k^*.$$
One verifies that $(f_1=f_2=0)$ is a non degenerate complete intersection in $\A^3(k)$ and the polyhedron of each has only one compact face which is a triangle. These triangles, say $\gamma_1,\gamma_2$, are not parallel so that there is no $\check a\in \R^d_{>0}$ which takes its minimum on both triangles. If we choose $x_0,y_0,z_0$ such that $x_0^2=1,y_0^2=3,z_0^3=-4$, and choose $\eta =\frac{2}{3}y_0^{-1}, \zeta=\frac{3}{4}z_0^{-1}$ then $f_{1,\gamma_1}=f_{2,\gamma_2}=0$ has $(x_0,y_0,z_0)$ as a singular point in the torus. So for the statement of the exercise to be true in this case one has to add to the non-degeneracy of $f_1,f_2$ an extra condition which ensures the transversality of $f_{1,\gamma_1}=0$ and $f_{2,\gamma_2}=0$ in the torus for pairs $\gamma_1,\gamma_2$ of compact faces which are not minimizers for some common $\check a$. This can be understood in terms of the positions of the strict transforms of $f_1,f_2$ after a desingularizing toric map. See \S 5 in the text.
\item Popescu Pampu suggested that in addition to  \cite{Hod} I should have added to the bibliography \cite{B1}, \cite{B2}, \cite{D1}, \cite{D2}.


\end{itemize}
%\begin{section}{Introduction}

%\end{section}
\begin{thebibliography}{7}

\bibitem{B1} H. F. Baker, {\em Examples of the application of Newton's polygon to the theory of singular points of algebraic functions}, Trans. Camb. Phil. Soc., Vol. XV, part IV, 1893, 403-450.

\bibitem{B2} H. F. Baker, {\em The practical determination of the deficiency (Geschlecht) and $\varphi$-curves for a Riemann surface}, Math. Ann., \textbf{45}, 1894, 118-132.

\bibitem{D1} G. Dumas. {\em Sur la r\'esolution des singularit\'es des surfaces}, C.R.A.S. Paris, t. 151, 1911, p. 682.

\bibitem{D2} G. Dumas. {\em Sur les singularit\'es des surfaces}, C.R.A.S. Paris, 1912, t. 154, no. 23, p. 1495.


\bibitem{Hod} W.V.D. Hodge, {\em The isolated singularities of an algebraic surface}, Proc. London Math. Soc. \textbf{30} (1930), 133-143.

\bibitem{Te}
B. Teissier, Monomial ideals, binomial ideals, polynomial ideals, in \textit{Trends in Commutative Algebra}, MSRI Publications no. 51, 2004, 211-246.
\end{thebibliography}

\end{document}
