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 \title{A correction in the statement of my theorem
on the efficient smooth cover and smooth stratification of an algebraic
variety}
\author{Alexander L.~Chistov%
 \\[2ex]
St.~Petersburg Institute for Informatics and
Automation of the\\ Academy of Sciences of Russia\\
14th line 39, St.~Petersburg 199178, Russia,\\
e-mail: labta@iias.spb.su  or sliss@iias.spb.su }
\date{\Large 2004}

\newtheorem{thms}{THEOREM}
\newtheorem{lems}{LEMMA}
\newtheorem{rems}{REMARK}
\newtheorem{defns}{DEFINITION}
\newtheorem{props}{PROPOSITION}
\newtheorem{coros}{COROLLARY}

\maketitle

\begin{abstract}
We correct the statement of our theorem about the construction of the
efficient smooth cover and smooth stratification of an algebraic variety in
zero--characteristic. Namely, this correction is related to the estimations
of the lengths of the coefficients from the ground field of the polynomials
giving the strata. The proof of this theorem, the bounds for degrees of
the strata and all the consequences of the theorem are without changes.
\end{abstract}

\newpage

In \cite{1} Theorem~2 we prove the existence of an
efficient smooth cover and smooth stratification an algebraic variety in
zero--characteristic.
Assertion (vi) of Theorem~2 from \cite{1} must be corrected as follows:
\begin{itemize}
\item[(vi)] {\it For all $\alpha\in A$, $1\le j\le s(\alpha)$
the lengths of coefficients from $k$ of polynomials $h_{\alpha,j}$
 are bounded from above
by a polynomial in
$n^{2^{s(\alpha)^{C}}}d^{ns(\alpha)}$, $d_1$, $d_2$, $M$, $M_1$, $m$
for an absolute constant $0<C\in {\Bbb R}$.
Further, in the case of smooth stratification the
lengths of coefficients from $k$ of all polynomials
of the family $f$ are bounded
from above by a polynomial in
$2^{2^{n^{C}}}d^{n^2}$, $d_1$, $d_2$, $M$, $M_1$, $m$
for an absolute constant $0<C\in {\Bbb R}$.}
\end{itemize}
Thus, one needs to replace $d^n$ by $d^{ns(\alpha)}$
for the case of smooth cover (respectively by $d^{n^2}$ for the case of the
smooth stratification)
in (vi) Theorem~2 \cite{1}.
All the other assertions of
this theorem and its proof are without changes. In particular, the degrees
of all the
strata are bounded from above by $2^{2^{n^C}}d^n$. The proof is the same
since actually
we did not estimate the lengths of coefficients from $k$ of these
polynomials explicitly.
The bound from (vi) is obtained straightforwardly by the recursive
application of Theorem~1 \cite{1},
see the proof Theorem~2 \cite{1}.

This correction implies the same correction in the same
assertion (vi) of Theorem~4
from \cite{2}. The working time
of the algorithm from Theorem~4
\cite{2} for constructing the smooth cover
(respectively the smooth stratification) is polynomial in the
size of the output and
$2^{2^{n^{C}}}d^n$, $d_1$, $d_2$, $M$, $M_1$, $m$
(respectively is polynomial
in $2^{2^{n^{C}}}d^{n^2}$, $d_1$, $d_2$, $M$, $M_1$, $m$),
where $0<C\in {\Bbb R}$ is an absolute constant.
So in the case of the smooth cover
we need to add here ``in the size of the output'' which might be
polynomial also $d^{n^2}$ by (vi).

Notice that the proof of Theorem~3 \cite{1} about computing the dimension of
a real algebraic variety is the same. It uses only the fact
that the degrees of the strata
from Theorem~2 \cite{1} are bounded from above by $2^{2^{n^C}}d^n$.
But now one can not use immediately Theorem~4 \cite{2} to deduce Theorem~3
\cite{1}.
So I know only one proof of Theorem~3 \cite{1}
using the original construction from \cite{1}.





 \begin{thebibliography}{88}

\bibitem{1} {\bf  Chistov A. L.:}
{\it ``Efficient Smooth Stratification of an Algebraic Variety in
Zero--Characteristic  and its Applications},
Zap. Nauchn. Semin. St-Petersburg.
Otdel. Mat. Inst. Steklov (POMI) v. 266 (2000) p.254--311 (in Russian)

\bibitem{2} {\bf  Chistov A. L.:} {\it
``Polynomial--time algorithms in zero--characteristic for a new
 model of representation of algebraic varieties''},
Preprint of the St.Petersburg
Mathematical Society (2004) \#10, http://www.MathSoc.spb.ru/preprints/

\end{thebibliography}

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