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\begin{document}

 \title{Efficient Construction of Local Parameters
 of Irreducible Components of an Algebraic Variety
 in Nonzero Characteristic}
\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 2005}

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

\maketitle

\begin{abstract}
Consider an $(n-s)$-dimensional algebraic variety $W$ defined over an
infinite field $k$ of nonzero characteristic $p$ and irreducible over this
field. Let $W$ be a subvariety of the projective space of dimension $n$.  We
prove that the local ring of $W$  has a sequence of local parameters
represented by $s$ non--homogeneous polynomials with the product of degrees
less than the degree of the variety multiplied to a constant depending on
$n$. This allows to prove the existence of effective smooth cover and smooth
stratification of an algebraic variety in the case of ground field of
nonzero characteristic. The paper extends the analogous results of the author
obtained earlier from zero to nonzero characteristic of the ground field.
\end{abstract}




\newpage

\section*{Introduction}

 Let $k$ be an infinite field
of characteristic $p$. Let ${\Bbb P}^n(\overline{k})$
be the $n$--dimensio\-nal
projective space over algebraic closure $\overline{k}$ of the field $k$
with the homogeneous coordinates $X_0,\ldots ,X_n$.
Let $W\subset{\Bbb P}^n(\overline{k})$ be an algebraic variety
defined over $k$ and irreducible over $k$. Suppose that
the dimension $\dim W=n-s$ where $1\le s\le n$ and the
degree $\deg W=D$. Let the variety $W$ be the set of all zeroes of the
homogeneous prime ideal ${\mathfrak P}\subset k[X_0,\ldots ,X_n]$.
In \cite{2} we prove that the local ring
of $W$  has a regular sequence (respectively if $p=0$ a sequence of local
parameters)
represented by $s$ non--homogeneous polynomials with the product
of degrees bounded from above by $D$ multiplied
to a constant depending on $n$ (this is proved in \cite{2}
only for the case of
perfect ground field; but from here  in the obvious way the same assertion
follows in the general case, see below).
Further, see \cite{2}, in the case when $W$ is an irreducible
component of an algebraic variety
given by a system of homogeneous polynomial equations of
degrees less than $d$ over a field of zero--characteristic,
the degrees of all these local parameters are less then $d$
multiplied to a constant depending on $n$.
These constants depending on $n$ are
estimated in \cite{2}.

The aim of the present paper is to extend the results of \cite{2}
related  to the bounds on the degrees of local
parameters of $W$
to the case of nonzero characteristic. Namely for arbitrary $p$ we prove
that $W$  has a sequence of local parameters
represented by $s$ non--homogeneous polynomials with the product
of degrees bounded from above by $D$ multiplied
to a constant depending only
on $n$, see Theorem~1. In the case when $W$ is an
irreducible component of an algebraic variety
given by a system of homogeneous polynomial equations of
degrees less than $d$, and $W$ is defined over $k$ and irreducible over $k$
the degrees of all these local parameters are less than $d$
multiplied to a constant depending only on $n$, see Theorem~2.
These constants depending on $n$ are
estimated.

Denote by $k^{p^{-\infty}}$ the perfect closure of $k$.
For
every integer $\nu\ge 0$ denote
by $k^{p^{-\nu}}$ the subfield of all the elements $z\in k^{p^{-\infty}}$
such that  $z^{p^\nu}\in k$.

Notice here that in the general case one can represent an algebraic variety
$V$ given by a system of homogeneous polynomial equations
with coefficients from $k$ of degrees less than $d$ only as a union of
defined
over $k^{p^{-\infty}}$ and irreducible over $k^{p^{-\infty}}$ components.
By \cite{1} actually
\begin{itemize}
\item[(*)] for every $0\le s\le n$
all the irreducible components of $V$ of codimension $s$ defined over
$k^{p^{-\infty}}$ are defined
also over $k^{p^{-\nu}}$ for an integer $\nu\ge 0$ such that $p^{\nu}\le
d^{2s}$.
\end{itemize}
Hence if $k$ is perfect the variety $V$ is represented as a union of
defined over $k$ and irreducible over $k$ components.


Now using Theorem~1 and Theorem~2 we get an analog of Theorem~2
\cite{3}. Recall the last result from \cite{3}.
Consider a projective algebraic variety $V$ which
is given as a set of common zeros of a family of homogeneous polynomials of
degrees less than $d$ in $n+1$ variables with coefficients from a field $k$
of zero--characteristic. In \cite{3} we prove that $V$ can be represented as
a union (respectively disjoint union) of at most $C(n)d^n$ (respectively
$C(n)d^{n(n+1)/2}$) smooth quasiprojective algebraic varieties with degrees
(herewith the degree of a quasiprojective algebraic variety is degree of its
closure in the Zariski topology) bounded from above by $C(n)d^n$ where
$C(n)<2^{2^{n^C}}$ for an absolute constant $C>0$.
A minor correction of assertion (vi) of Theorem~2 \cite{3} see in \cite{4}
(it is related to the estimations
of the lengths of the coefficients from the ground field of the polynomials
giving the strata).
By Remark~4 from \cite{3}
at present the result similar to Theorem~2 \cite{3} is also true for the case
of the perfect ground
field of nonzero characteristic. In Theorem~3 below we consider even
the case when $k$ is not necessarily perfect.



For arbitrary polynomials $g_1,\ldots , g_m\in \overline{k}[X_0,\ldots ,
X_n]$ we shall denote by
${\cal Z}(g_1,\ldots , g_m)$ the set of all common zeroes
of these polynomials in ${\Bbb P}^n(\overline{k})$ (in what follows we
shall use the similar notations for the sets
of common zeroes of others polynomials or polynomial ideals
in projective or affine spaces).
For an arbitrary algebraic variety $V\subset{\Bbb P}^n(\overline{k})$
we shall denote by
${\cal I} (W)\subset k[X_0,\ldots ,X_n]$ the homogeneous ideal
corresponding to $W$ (the similar notations will be used for ideals
of affine algebraic varieties and for ideals in the rings of polynomials over
other fields of coefficients).

Recall some definitions from \cite{2}.

\par\medskip\noindent{\bf DEFINITION~1}\hspace{0.1em} {\it  Let $W$ and
${\mathfrak P}$ be as above.
Let $0<j\le s$ be an integer. Consider a sequence of
homogeneous polynomials
$f_i\in {\mathfrak P}$, $1\le i\le j$.
Let $0\le u<j$ be an integer. Denote by
${\cal W}={\cal W}({\mathfrak P},u)$ the set of all
defined over $k$ and irreducible over $k$
components $E$ of the algebraic variety
${\cal Z}(f_1,\ldots , f_u)$ such that $E\supset W$.
Suppose that
for every integer $0\le u< j$
the polynomial $f_{u+1}\in {\mathfrak P}$
is a polynomial $g$ of the least
degree such that for every irreducible
component $E\in {\cal W}$ the
algebraic variety
$$
E\not\subset {\cal Z}(g).
$$
Then any such sequence $f_1,\ldots , f_j$ will be called
${\mathfrak P}$--sequence (of length $j$).
By Proposition~1, see below, there exists a ${\mathfrak P}$-sequence
$f_1,\ldots,f_s$.
}\par\medskip



\par\medskip\noindent{\bf DEFINITION~2}\hspace{0.1em} {\it
Let $f_1,\ldots , f_s$ be a ${\mathfrak P}$--sequence. By definition put
$$
\delta_i({\mathfrak P})=\deg f_i, \quad 1\le i\le s .
$$
Then, see \cite{2}
Lemma~2 Section~2, the integers $\delta_i({\mathfrak P})$ depend only on
${\mathfrak P}$ and do not depend of the choice of the ${\mathfrak
P}$--sequence. The integer $\delta_i({\mathfrak P})$ is called the $i$--th
degree
corresponding to ${\mathfrak P}$--sequence.
}\par\medskip


Note that if $f_1,\ldots , f_j$ is a
${\mathfrak P}$--sequence then for every  $0\le u\le j$
every irreducible component of the algebraic variety
${\cal Z}(f_1,\ldots , f_u)$ containing $W$ is of dimension $n-u$.
In particular when $j=s$ the algebraic variety $W$
is an irreducible component of ${\cal Z}(f_1,\ldots , f_s)$.
Note also that in the general
case the algebraic variety ${\cal Z}(f_1,\ldots , f_s)$ may have
irreducible components of dimension greater than $n-s$.

The following result follows from Theorem~1 \cite{2}.

\par\medskip\noindent{\bf PROPOSITION~1}\hspace{0.1em} {\it  Let $k$ be an
infinite field. Then there exists some
${\mathfrak P}$--sequence
$f_1,\ldots , f_s$. Hence
$\delta_i=\delta_i({\mathfrak P})$
$=\deg f_i$  for every $1\le i\le s$. Further, see \cite{2}
Theorem~1, for arbitrary $p$
there is $C\in{\Bbb R}$ such that the inequality
$$
\delta_1\cdot\ldots \cdot\delta_s\le n^{2^{s^{C}}}D
\eqno (1)
$$
holds.
}\par\medskip

\noindent{\bf PROOF}\quad The case when $k$ is perfect is exactly Theorem~1
\cite{2}.
In the general case we extend $k$ till $k^{p^{-\infty}}$ and get a
${\mathfrak P}$--sequence $f_1,\ldots , f_s$ with coefficients from
$k^{p^{-\infty}}$.
Since $W$ is defined over $k$ and irreducible over $k$, one can represent
$f_i=\sum_{1\le j\le N_i}
\lambda_{i,j}g_{i,j}$, where all \(\lambda_{i,j}\in k^{p^{-\infty}}\)
and $g_{i,j}\in{\mathfrak P}$ are
nonzero homogeneous polynomials (with coefficients from $k$), and $\deg
g_{i,j}=\deg f_i$ for all $i,j$.
Therefore, $W$ is an irreducible component of the algebraic variety
$$
{\cal Z}\Bigl(\,\sum_{1\le j\le N_1}\mu_{1,j}g_{1,j},\ldots ,
\sum_{1\le j\le N_s}\mu_{s,j}g_{s,j}\,\Bigr)
$$
for appropriate $\mu_{i,j}\in k$, $1\le i\le s$, $1\le j\le N_i$.
This gives the required ${\mathfrak P}$-sequence with coefficients from $k$.
The proposition is proved.

\par\medskip\noindent{\bf REMARK~1}\hspace{0.1em} {\it
Note that the proof of the fact that  $\delta_i({\mathfrak P})$
does not depend on the choice
of ${\mathfrak P}$--sequence uses the infiniteness of $k$.
Theorem~1 \cite{2} is valid also for the case of finite fields
(but it is not affirmed here that $\delta_i$ does not depend on the choice
of ${\mathfrak P}$--sequence). One can easily see
that only minor changes are required in the proof. But we don't consider
now the most general situation.
}\par\medskip



Now we are going to formulate the results of the present paper.
Theorem~1 and Theorem~2, see below, generalize Theorem~2 and
Theorem~3
from \cite{2} to the case of nonzero characteristic.

\par\medskip\noindent{\bf THEOREM~1}\hspace{0.1em} {\it
Let $k$ be an infinite field of arbitrary characteristic $p$.
Let $W={\cal Z}({\mathfrak P})$ and ${\mathfrak P}$ be as
above. Then there are homogeneous polynomials
$g_1,\ldots ,g_s\in {\mathfrak P}$,
a smooth point point $x\in W$, and a linear form
$L\in k[X_0,\ldots ,X_n]$
 such that
\begin{itemize}
\item $L(x)\ne 0$,
\item $g_1/L^{\deg g_1},\ldots , g_s/L^{\deg g_s}$ is a system of local
parameters (definition of local parameters, see, e.g., in \cite{6}),
of the algebraic variety $W$ at the point $x$,
\item $\deg g_i\le n^{2^{s^{C}}}\delta_i({\mathfrak P})$, $1\le i\le s$,
for an absolute constant $0<C\in{\Bbb R}$ (it does not depend on $p$).
\end{itemize}
}\par\medskip



As a consequence of Theorem~1 and \cite{2} Lemma~5 we get the
following result.
\par\medskip\noindent{\bf THEOREM~2}\hspace{0.1em} {\it
Let $\phi_1,\ldots ,\phi_m\in k[X_0,\ldots ,X_n]$, $m\ge 1$, be
arbitrary homogeneous polynomials.
Let the degrees $\deg \phi_i< d$ for all $1\le i\le m$.
Assume that there is a defined over $k$ and irreducible over $k$ component
$W$ of the algebraic variety
$$
{\cal Z}(\phi_1,\ldots ,\phi_m)\subset {\Bbb P}^n(\overline{k})
$$
of all common zeroes of these polynomials in
${\Bbb P}^n(\overline{k})$. Let $\dim W=n-s$ where $1\le s\le n$.
Then there are homogeneous polynomials
$g_1,\ldots ,g_s\in {\mathfrak P}$,
a smooth point point $x\in W$, and a linear form
$L\in k[X_0,\ldots ,X_n]$
 such that
\begin{itemize}
\item $L(x)\ne 0$,
\item $g_1/L^{\deg g_1},\ldots , g_s/L^{\deg g_s}$ is a system of local
parameters
of the algebraic variety $W$ at the point $x$,
\item $\deg g_i\le n^{2^{s^{C}}}d$, $1\le i\le s$,
for an absolute constant $0<C\in{\Bbb R}$ (it does not depend on $p$).
\end{itemize}
}\par\medskip


Now we are going to formulate an analog of Theorem~2 \cite{3}.
But at first we need to recall some definitions from \cite{3}.
Let $f_1,\ldots ,f_m\in k[X_0,\ldots ,X_n]$, $m\ge 1$, be
arbitrary homogeneous polynomials (one should not confuse theì
with
the ones from the Definition~1, Definition~2 and Proposition~1).
Let the degrees $\deg f_i< d$ for all $1\le i\le m$.
Put $V={\cal Z}(f_1,\ldots , f_m)\subset{\Bbb P}^n(\overline{k})$
to be the algebraic variety of all the common zeroes of the polynomials
$f_1,\ldots ,f_m$ in the projective space ${\Bbb P}^n(\overline{k})$.

\par\medskip\noindent{\bf DEFINITION~3}\hspace{0.1em} {\it  Smooth cover of
the algebraic variety $V$ is a finite family
$$
V_\alpha,
\quad\alpha\in A,\eqno (2)
$$
of quasiprojective smooth
algebraic varieties $V_\alpha\subset{\Bbb P}^n(\overline{k})$, $\alpha\in
A$ such that $V$ is represented as a union
$V=\cup_{\alpha\in A}V_\alpha$. Further, we shall require that
all irreducible components of $V_\alpha$ have the same dimension (which
depends only on $\alpha$).  Smooth stratification of the algebraic
variety $V$ is a smooth cover $V_\alpha$,
$\alpha\in A$, of $V$ such that additionally
for any two $\alpha_1,\alpha_2\in A$ if
$\alpha_1\ne \alpha_2$ then $V_{\alpha_1}\cap V_{\alpha_2}=\emptyset$.
}\par\medskip


In \cite{3} we introduce a new method to represent algebraic varieties.
Only the case of zero--characteristic is considered in \cite{3}.
Now we extend this representation for nonzero characteristic.
In the statement of Theorem~3 below the ground field $k$ is
finitely generated over a
primitive subfield (hence it is not necessarily perfect).
Namely we represent $k={\Bbb F}_p(t_1,\ldots ,t_l,\theta)$,
$l\ge 0$, where $t_1,\ldots ,t_l$
are algebraically independent over the field of
$p$ elements ${\Bbb F}_p$, and $\theta$ is algebraic and separable over
the field
${\Bbb F}_p(t_1,\ldots ,t_l)$ element. The minimal polynomial
$F\in{\Bbb
F}_p(t_1,\ldots ,t_l)[Z]$ of the element $\theta$ over
${\Bbb F}_p(t_1,\ldots ,t_l)$ is given. The
leading coefficient $\mbox{\rm lc}_ZF$ of the polynomial $F$ with
respect to $Z$ is $1$. In the case when $k$ is finite, i.e., $l=0$, we shall
require
that $k$ has sufficiently many elements. Namely, the number of elements
$\#k$ is bounded from below
by a polynomial in
$2^{2^{n^{C}}}d^{n^2}$ for an appropriate $C>0$ (where $\deg f_i<d$, see
above).
One can always extend the finite ground field till a field $k(t_1)$,
or, in the other way, till
a finite field $k'$ with sufficient many elements, and after that apply
Theorem~3.




We represent each polynomial $f=f_i$ in the form
$$
 f=\frac{1}{a_0} \sum_{i_0, \ldots, i_n} \sum_{0 \leq j <
 \deg_Z F} a_{i_0, \ldots, i_n,j} \theta^j X_0^{i_0} \cdots X_n^{i_n}
 \, ,
$$
 where $a_0,a_{i_0, \ldots, i_n,j} \in{\Bbb F}_p[t_1, \ldots, t_l], \,
\mbox{\rm G\,C\,D\,}_{i_0, \ldots, i_n,j} (a_0,a_{i_0, \ldots,
i_n,j})=1$.
 Let us define the length $l(a)$ of an element $a\in{\Bbb F}_p$ by the
formula
 $l(a)=[\log p/\log 2]+1$.
 Similarly the length of coefficients $l(f)$ of the polynomial $f$ is
defined to
 be $l(f)=[\log p/\log 2]+1$ and the degree
$$
 \deg_{t_\gamma} (f)=\max_{i_0, \ldots, i_n,j} \{\deg_{t_\gamma} (a_0),
 \deg_{t_\gamma} (a_{i_0, \ldots, i_n,j})\} \, ,
$$
 where $1 \le\gamma\le l$.
 In the similar way we shall define degrees and lengths
of coefficients from ${\Bbb F}_p$ of other polynomials,
in particular $\deg_{t_\gamma} F$ and $l(F)=[\log p/\log 2]+1$ are defined.

 \noindent
 We shall suppose that we have the following bounds
 \begin{eqnarray*}
 \deg_{X_0, \ldots, X_n} (f_i)<d, \; \deg_{t_\gamma}(f_i)<d_2, \\
 \deg_Z (F)<d_1, \; \deg_{t_\gamma} (F)<d_1.
 \end{eqnarray*}
 The size $L(f)$ of the polynomial $f$ is defined to be the product of
 $l(f)$ to the number of all the coefficients from ${\Bbb Z}$ of $f$ in
 the dense representation.
 We have
$$
 L(f_i)<({d+n \choose n}d_1+1)d_2^l([\log p/\log 2]+1).
$$
 Similarly $L(F)<d_1^{l+1} ([\log p/\log 2]+1)$.  Unless
 we state otherwise, in what follows we suppose $l$ to be fix.


Actually, in \cite{3}
only minor modifications are required.
The first of them is related to the estimation of the degrees of
inseparability of the
fields of definition of the strata. The second one is connected to the
choice of coefficients
of linear forms and other polynomials, see below. In place of integer
coefficients we choose the coefficients from a small
subring  $\Lambda$
of the field $k$.
If the transcendency degree of $k$ over
the primitive subfield ${\Bbb F}_p$
of $p$ element is nonzero,
then we choose an element $\lambda^{(0)}\in k$
which is transcendental over
${\Bbb F}_p$ (say $\lambda^{(0)}=t_1$) and
put $\Lambda={\Bbb F}_p[\lambda^{(0)}]\subset k$. If the transcendency
degree of $k$ over ${\Bbb F}_p$ is zero,
then set $\Lambda=k$ (recall that in this case we assume that
$k$ has sufficiently many elements, see above).
The size
${\rm L}(a)$ of an arbitrary
element $a\in\Lambda$ is defined, see above (e.g.,
if $\Lambda={\Bbb F}_p[\lambda^{(0)}]$ then
${\rm L}(a)=(\deg_{\lambda^{(0)}}(a)+1)([\log p/\log 2]+1)$).

Let $W$ be a quasiprojective algebraic variety in ${\Bbb
P}^n(\overline{k})$ and $W$ is defined over $k$.
Then represent
$$
W=W^{(1)}\setminus \cup_{2\le i\le a}W^{(i)},\eqno (3)
$$
where
all $W^{(i)}$, $1\le i\le a$ are projective algebraic varieties in ${\Bbb
P}^n(\overline{k})$ defined over $k$.
Each algebraic variety $W^{(i)}$, $1\le i\le a$
is given as a union of some irreducible components of the
algebraic variety ${\cal Z}(f^{(i)}_1,\ldots , f^{(i)}_{m(i)})\subset
{\Bbb P}^n(\overline{k})$, where homogeneous polynomials
$f^{(i)}_1,\ldots , f^{(i)}_{m(i)}\in k[X_0,\ldots ,X_n]$ are given.
For every $1\le s\le n$
denote by $V^{(i)}_s$ (respectively $W^{(i)}_s$)
the union of all irreducible components of dimension $n-s$
of ${\cal Z}(f^{(i)}_1,\ldots , f^{(i)}_{m(i)})$ (respectively
$W^{(i)}$).
Therefore, $W^{(i)}_s$ is a union of
some irreducible components of $V^{(i)}_s$.
For every $1\le s\le n$ the
family of linear forms $L^{(i)}_{s,s+1},\ldots , L^{(i)}_{s,n}\in
k[X_0,\ldots ,X_n]$  with
coefficients from $\Lambda$ is given such that the number of points
$$
 \#V^{(i)}_s\cap{\cal Z}(L^{(i)}_{s,s+1},\ldots , L^{(i)}_{s,n})
<+\infty \eqno (4)
$$
is finite. Every point
$\xi\in V^{(i)}_s\cap{\cal Z}(L^{(i)}_{s,s+1},\ldots , L^{(i)}_{s,n})$
is a smooth point
of the algebraic variety ${\cal Z}(f^{(i)}_1,\ldots , f^{(i)}_{m(i)})$,
and
the intersection of the tangent spaces in the point $\xi$ of $V^{(i)}_s$ and
${\cal Z}(L^{(i)}_{s,s+1},\ldots , L^{(i)}_{s,n})$ is transversal, i.e.,
$$
 T_{\xi, V^{(i)}_s}\cap
{\cal Z}(L^{(i)}_{s,s+1},\ldots , L^{(i)}_{s,n})=\{\xi\}.\eqno (5)
$$
The set of points
$$
\Xi^{(i)}_{s}= W^{(i)}_s
\cap{\cal Z}(L^{(i)}_{s,s+1},\ldots , L^{(i)}_{s,n})
$$
is given. Hence the following property holds.
Let $\xi\in V^{(i)}_s
\cap{\cal Z}(L^{(i)}_{s,s+1},\ldots , L^{(i)}_{s,n})$
and $V'$ be the uniquely defined irreducible over $k$
component
of the algebraic variety $V^{(i)}_s$ such that $\xi\in V'$. Then $\xi\in
\Xi^{(i)}_{s}$ if and only if $V'$ is a component of $W^{(i)}$.

Thus, formally the accepted in this paper representation of
$W$ is a triple
$$
(f,L,\Xi),\eqno (6)
$$
where $f$ is a
family of polynomials
$$
 f^{(i)}_{j},\quad 1\le j\le m(i),\,1\le i\le a, \eqno (7)
$$
$L$ is a family of linear forms
$$
 L^{(i)}_{s,j},\quad s+1\le j\le n,\,  1\le s\le n,\, 1\le i\le a,
\eqno (8)
$$
and $\Xi$ is a family of finite sets of points
$$
 \Xi^{(i)}_{s},\quad 1\le s\le n,\, 1\le i\le a. \eqno (9)
$$
Denote also
$$
\Xi^{(i)}=\bigcup_{1\le s\le n}\Xi^{(i)}_{s}.
\eqno (10)
$$
Note that here $W$ is defined over $k$. We shall use also this
representation for other fields of definition.

Now we are able to formulate the analog of Theorem~2 \cite{3}.
\par\medskip\noindent{\bf THEOREM~3}\hspace{0.1em} {\it  Let the ground
field $k={\Bbb F}_p(t_1,\ldots ,t_l,\eta)$ be
given as above. Let homogeneous polynomials $f_1,\ldots ,f_m\in k[X_0,\ldots
, X_n]$ and the algebraic variety
$V={\cal Z}(f_1,\ldots ,f_m)$ be as above.
There is a smooth cover
(respectively a smooth stratification) $V_\alpha$,
$\alpha\in A$, of
the algebraic variety $V$ such that every
quasiprojective algebraic variety $V_\alpha$
is defined over $k^{p^{-n(\alpha)}}=k_\alpha$
for an integer $n\ge 0$, irreducible over
$k_\alpha$, and
represented in the accepted
way (with the ground field $k_\alpha$ in place of $k$).  Let
$\dim V_\alpha=n-s$ where $1\le s\le n$, and $s=s(\alpha)$
depends on $\alpha$.
Let (6) be the constructed representation of $V_\alpha$.
Denote
$h_{\alpha,j}=f^{(1)}_{j}$, $1\le j\le m(1)$, and
$\Delta_\alpha=f^{(2)}_{1}$ if $a\ge 2$, see (7).
Then the constructed representation of $V_\alpha$ satisfies the following
properties.
\begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})}
\item For the case of smooth cover $a=2$ if $s<n$, and $a=1$ if $s=n$.
For the case of smooth stratification $a\le 2^{2^{n^{C}}}d^{n(n+1)/2}$
for an absolute constant $0<C\in {\Bbb R}$.
\item $m(1)=s$ and if $a\ge 2$ then $m(2)=1$. Hence if $a\ge 2$ then
$V_\alpha$ is an irreducible component of the algebraic variety
${\cal Z}(h_{\alpha,1},\ldots , h_{\alpha,s})
\setminus{\cal Z}(\Delta_\alpha)$ in the
case of the smooth cover
(respectively an open in the Zariski topology subset of
an irreducible component of the latter algebraic
variety in the case of smooth stratification).
\item There are linearly independent linear forms $Y_0,\ldots , Y_n\in
k[X_0,\ldots ,X_n]$ such that
$X_i=\sum_{0\le j\le n}x_{i,j}Y_j$, all the coefficients
$x_{i,j}\in k$,  all $x_{i,j}$ are elements from $\Lambda$ with lengths
$O(2^{n^C}+n\log d)$
for an absolute constant $C>0$, and
$$
\Delta_\alpha=
\mbox{\rm det}  (\partial h_{\alpha,i}/\partial Y_j)_{1\le i,j\le
s}=
\mbox{\rm det}\Bigl(\,\sum_{0\le v\le n}x_{v,j}
\partial h_{\alpha,i}/\partial X_v\,\Bigr)_{1\le i,j\le s}.
$$
Hence $V_\alpha$ is a smooth algebraic variety by the
implicit function theorem.
Besides that, in the case of smooth cover
one can take $Y_i=X_{\sigma(i)}$ for some
permutation $\sigma$ of the set $0,\ldots , n$.
\item The lengths of coefficients from $\Lambda$ of all linear forms from the
family $L$ is $O(2^{n^C}+n\log d)$
for an absolute constant $0<C\in {\Bbb R}$.
\item For all $\alpha\in A$,
$1\le j\le s(\alpha)$
the degrees $\deg_{X_0,\ldots , X_n}h_{\alpha,j}$
are less than $n^{2^{s(\alpha)^{C}}}d$
for an absolute constant $0<C\in {\Bbb R}$.
In the case of smooth stratification for all $i>2$, $j$
degrees $\deg_{X_0,\ldots , X_n}f^{(i)}_{j}$
are less than $2^{2^{n^{C}}}d$.
\item For all $\alpha\in A$, $1\le j\le s(\alpha)$
the polynomials
$$
h_{\alpha,1},\ldots , h_{\alpha,s(\alpha)}\in
k^{p^{-\nu(\alpha)}}[X_0,\ldots
, X_n],
\eqno (11)
$$
where $p^{\nu(\alpha)}\le n^{2^{s(\alpha)^{C}}}d^{2ns(\alpha)}$
for an absolute constant $0<C\in {\Bbb R}$. Further,
the lengths of coefficients of polynomials
$h_{\alpha,j}^{p^{\nu(\alpha)}}$ 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 $0<C\in {\Bbb R}$.
Finally, in the case of smooth stratification
$p^{n(\alpha)}\le 2^{2^{n^{C}}}d^{2n^2}$
for an absolute constant $0<C\in {\Bbb R}$;
for all $i>2$, $j$ the lengths of coefficients of all polynomials
from the family $(f_j^{(i)})^{p^{n(\alpha)}}$,  $1\le j\le m(i)$,
$1\le i\le a$, see (7), 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 $0<C\in {\Bbb R}$.
\item The number of elements $\# A$ of $A$ is bounded from above by
$2^{2^{n^{C}}}d^{n}$ for the case of smooth cover
(respectively $2^{2^{n^{C}}}d^{n(n+1)/2}$
for the case of smooth stratification)
for an absolute constant $0<C\in {\Bbb R}$.
\end{enumerate}
}\par\medskip

\noindent{\bf SKETCH OF THE PROOF}\quad This follows from Remark~4 \cite{3}.
Namely, applying Theorem~1 and Theorem~2 in place of Theorem~1 \cite{3}
(we do not need at present to construct regular sequences and sequences of
local parameters as in assertions (i) and (ii) of the last theorem,
and we use only the existence of them)
the reader can repeat all the constructions from the proof of Theorem~2
\cite{3} with small modifications
and obtain all the assertions of
Theorem~3.

Let us consider in more details assertion (vi). It is sufficient here
to prove only two first statements related  to the polynomials
$h_{\alpha,1},\ldots, h_{\alpha,s(\alpha)}$. Indeed,
if they are proved for strata of all dimensions then
the last assertion from (vi) follows immediately from (ii).

The statements related to the polynomials
$h_{\alpha,1},\ldots, h_{\alpha,s(\alpha)}$ are proved by the
induction on $s(\alpha)$ using the construction from the proof of Theorem~2
\cite{3}. Here one should use (*), see above.
We leave the details to the reader.
The theorem is proved.








\section{Preliminary definitions and lemmas}\label{s1}

In the present and the next sections
our aim is to prove Theorem~1. We shall follow the construction from
Section~10 \cite{2} and extend it to the case of nonzero characteristic.
Since the case of zero--characteristic is considered in \cite{2}, we shall
assume in what follows without loss of generality that $p=\mbox{\rm
char}(k)>0$. Besides that, cf. the proof of Proposition~1, we shall
assume without loss of generality that
the field $k$ is perfect.

Let the homogeneous prime ideal ${\mathfrak P}$ be the same as in
the Introduction, the algebraic variety
$W={\cal Z}({\mathfrak P})$ in ${\Bbb P}^n(\overline{k})$ and $D=\deg
W$. Let $s\ge 1$.
Let $Y_0,\ldots , Y_n\in k[X_0,\ldots ,X_n]$ be linearly independent linear
forms satisfying the
following conditions
\setcounter{equation}{11} \begin{eqnarray}
&&W\cap{\cal Z}(Y_0,Y_{s+1},\ldots , Y_n)=\emptyset, \label{12} \\
&&W\cap{\cal Z}(Y_0,Y_{i},Y_{s+1},\ldots , Y_{n-1})=\emptyset, \quad
1\le i\le s. \label{13}
\end{eqnarray}
Consider the projections
\begin{eqnarray*}
&&\pi\, :\, W
\rightarrow {\Bbb P}^{n-s}(\overline{k}),  \\
&&(X_0:\ldots : X_n)\mapsto (Y_0:Y_{s+1}:\ldots :Y_n),
\end{eqnarray*}
and
\begin{eqnarray*}
&&\pi_i\, :\, W\rightarrow {\Bbb P}^{n-s+1}(\overline{k}), \\
&&(X_0:\ldots : X_n)\mapsto (Y_0:Y_i:Y_{s+1}:\ldots :Y_n), \,
1\le i\le s.
\end{eqnarray*}
All the projections
$\pi$ and $\pi_i\, :\, W\rightarrow\pi_i(W)$, $1\le i\le s$,
are finite by (12). Let
$H_i\in k[Z_0, Z_s,Z_{s+1},\ldots , Z_n]$
be the form corresponding to the projection $\pi_i$
of $W$ for every $1\le i\le s$, see Definition~6 from Section~8 \cite{2},
i.e., it is a nonzero polynomial of the least degree such that
$H_i(Y_0,Y_i,Y_{s+1},\ldots , Y_n)$ vanishes on $W$.
The following lemma is similar to Lemma~17 \cite{2}.

\par\medskip\noindent{\bf LEMMA~1}\hspace{0.1em} {\it
Let $x\in W$ be a smooth point in ${\Bbb P}^n(\overline{k})$.
Then there is a family of linearly independent linear forms
$Y_0,\ldots , Y_n\in k[X_0,\ldots ,X_n]$
satisfying (12) and (13) and such that
\begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})}
\item $Y_0(x)\ne 0$,
\item $\pi^{-1}(\pi(x))$ consists of $\deg W$ different points,
\item the number of elements
$\# (Y_i/Y_0)(\pi^{-1}(\pi(x)))=\# \pi^{-1}(\pi(x))$ for every $1\le
i\le s$.
\end{enumerate}
Let $L\in k[X_0,\ldots ,X_n]$ be a linear form such that
$L(x)\ne 0$. Set
$$
\varphi_i=H_i(Y_0,Y_i,Y_{s+1},\ldots , Y_n)/L^{\deg H_i},
\quad 1\le i\le s.
$$
Then $\deg H_i=D$,
the polynomial $H_i\in k(Z_0,Z_{s+1},\ldots , Z_n)[Z_s]$ is separable for
every $1\le i\le s$  and
$\varphi_1,\ldots , \varphi_s$ is a system
of local parameters of the algebraic variety $W$ at the point $x$ (this
means by definition that
$\varphi_1,\ldots , \varphi_s$
generate the ideal of $W$ in the local ring ${\cal O}_{x,{\Bbb
P}^n(\overline{k})}$
of the point  $x$ in ${\Bbb P}^n(\overline{k})$).
}\par\medskip

\noindent {\bf PROOF} \quad
The existence of the linear forms $Y_0,\ldots , Y_n$ satisfying all the
requirements
follows from the fact that generic linear forms in $X_0,\ldots , X_n$
satisfy these requirements.
By (iii) and by, e.g.,
the remarks before the Definition~6 Section~8 from \cite{2}
the degree $\deg_{Z_s}H_i=\deg H_i=D=\deg W$. Hence
the polynomial $H_i\in k(Z_0,Z_{s+1},\ldots , Z_n)[Z_s]$ is separable.
By (ii) the differential $d_x\pi$ at the point $x$
of the projection $\pi$ is an isomorphism.
The implicit function theorem implies now $\varphi_1,
\ldots ,\varphi_s$ is a system of local parameters of $W$ at the point $x$.
The lemma is proved.

\medskip Suppose that a ${\mathfrak P}$--sequence $f_1,\ldots,f_s$ is given.
In this an the next sections we describe a construction
of local parameters at a point of the algebraic variety $W$.
We shall construct the polynomials $g_1,\ldots ,g_s$ from the
statement of Theorem~1.
Let $0\le r<s$ be an integer.
Suppose that we have already constructed by the recursion
on $r$ a smooth point $x\in W$ and homogeneous polynomials
$$
g_1,\ldots ,g_r,f_{r,r+1},\ldots , f_{r,s}\in k[X_0,\ldots ,X_n]
$$
such that
\begin{enumerate}\renewcommand{\labelenumi}{(\alph{enumi})}
\item $x$ is a smooth point of ${\cal Z}(g_1,\ldots ,g_r)$,
\item $W$ is an irreducible component of ${\cal Z}(g_1,\ldots
,g_r,f_{r,r+1},\ldots , f_{r,s})$,
\item $x$ is a smooth point of ${\cal Z}(g_1,\ldots
,g_r,f_{r,r+1},\ldots , f_{r,s})$,
\item $f_{r,j}=\sum_{1\le v\le j}q_{r,j,v}f_v$ where $q_{r,j,v}\in
k[X_0,\ldots ,X_n]$
are homogeneous polynomials and $\deg q_{r,j,v}=\deg f_j-\deg f_v$ for all
$1\le v\le j$.
\end{enumerate}
(so in the case of the base of the recursion $r=0$  a smooth point
$x\in W$ is given and we set $f_{0,j}=f_j$ for every $1\le j\le s$).
Our aim is to construct the polynomial $g_{r+1}$ and homogeneous
polynomials $f_{r+1,r+2},\ldots , f_{r+1,s}$
such that properties (a)--(d) will be satisfied
if one replaces $r$ by $r+1$ (may be with another point $x$).

Let $L=X_i$ for some $0\le i\le n$ and $L(x)\ne 0$.
Performing if necessary a permutation of coordinates we
shall assume without loss of generality that $L=X_0$.

Denote $\psi_i=g_i/X_0^{\deg g_i}$, $1\le i\le r$ and
$\rho_j=f_{r,j}/X_0^{\deg f_{r,j}}$, $r+1\le j\le s$.
Consider the system of local parameters $\varphi_1,\ldots , \varphi_s$ from
Lemma~1. By conditions (a)--(d)
there exist indices $1\le i_{r+1}<\ldots < i_s\le s$ such that
$$
\psi_1,\ldots ,\psi_r, \varphi_{i_{r+1}},\ldots , \varphi_{i_s}
$$
is a system of local parameters of $W$ at the point $x$. We shall suppose
without loss of generality that $i_j=j$
for all $r+1\le j\le s$.

By $Z_i=X_i/X_0-(X_i/X_0)(x)$, $1\le i\le n$, denote
the coordinate functions of the affine space
${\Bbb A}^n(\overline{k})={\Bbb P}^n(\overline{k})\setminus{\cal
Z}(X_0)$.
{\it In what follows for all polynomials
$w\in\overline{k}[Z_1,\ldots ,Z_n]$
(respectively $w\in\overline{k}[X_0,\ldots , X_n]$) the degree $\deg w$ is
the degree
with respect to $Z_1,\ldots,Z_n$ (respectively $X_0,\ldots , X_n$).}

So $\varphi_i,\psi_j, \rho_w\in \overline{k}[Z_1,\ldots ,Z_n]$ for all
$i,j,w$.
Let us choose indices $1\le i_{s+1}<\ldots< i_n\le n$ such that
$$
\psi_1,\ldots ,\psi_r, \varphi_{r+1},\ldots ,
\varphi_{s},Z_{i_{s+1}},\ldots
,Z_{i_n}
$$
is a system of local parameters of the algebraic variety $\{x\}$ consisting
of one point $x$.
We shall suppose without loss of generality  that $i_j=j$
for all $s+1\le j\le n$.

Effecting if necessary a permutation of coordinates $Z_1,\ldots , Z_s$ we
shall assume without loss of generality that
the minor of the Jacobi matrix at the point~$x$
$$
\Delta=
\det\left((\partial \psi_i/\partial Z_j\right)(x))_{1\le i,j\le r}\ne 0.
\eqno (14)
$$
Notice that $\Delta\in\overline{k}[Z_1,\ldots ,Z_n]$, and the degree
$\deg\Delta\le-r+\deg\psi_1+\ldots +\deg\psi_r$.


Consider the ring of formal power series
$$
\overline{k}[[\psi_1,\ldots ,\psi_r,
\varphi_{r+1},\ldots ,
\varphi_s,Z_{s+1},\ldots ,Z_n ]]=\Omega_0.
\eqno (15)
$$
This ring coincides with the ring of formal power series in $Z_1,\ldots
,Z_n$ over $\overline{k}$. Therefore
it contains the ring $k[X_1/X_0,\ldots ,X_n/X_0]$.
Each element $F$ of the ring (15) can be represented in  the form
$$
F=\sum_{i_1,\ldots , i_n\ge 0}F_{i_1,\ldots ,
i_n}\psi_1^{i_1}\cdot\ldots\cdot \psi_r^{i_r}
\varphi_{r+1}^{i_{r+1}}\cdot\ldots\cdot
\varphi_{s}^{i_{s}}Z_{s+1}^{i_{s+1}}\cdot\ldots\cdot
Z_{n}^{i_{n}},
\eqno (16)
$$
where all $F_{i_1,\ldots ,i_n}\in \overline{k}$.
Put
$$
\Omega=\overline{k}[[\psi_1,\ldots ,\psi_r,Z_{s+1},\ldots ,
Z_n]]\subset\Omega_0.
$$
Hence $\Omega$ is a subring of $\Omega_0$.

The differentiations $\partial/\partial\psi_i$, $1\le i\le r$, and
$\partial/\partial\varphi_i$, $r+1\le j\le s$,
are defined in the natural way on the ring $\Omega_0$.
The differentiations $\partial/\partial Z_i$, $1\le i\le s$ are defined in
the natural way on the ring
$\overline{k}[[Z_1,\ldots ,Z_n]]$. We define the differentiations
$\partial/\partial Z_i$, $1\le i\le s$, on $\Omega_0$
using the natural identification $\Omega_0=\overline{k}[[Z_1,\ldots ,Z_n]]$.

Set
\begin{eqnarray*}
&&\widetilde{\mathfrak P}=(\psi_1,\ldots ,\psi_r, \varphi_{r+1},\ldots ,
\varphi_s)\subset\Omega_0, \\
&&{\mathfrak q}=\widetilde{\mathfrak Q}=
(\psi_1^p,\ldots ,\psi_r^p, \varphi_{r+1}^p,\ldots ,
\varphi_s^p)\subset\Omega_0,\\
&&\widetilde{\mathfrak P}_r=(\psi_1,\ldots ,\psi_r)\subset\Omega_0,
\end{eqnarray*}
i.e., $\widetilde{\mathfrak P}$ (respectively
$\widetilde{\mathfrak Q}$, $\widetilde{\mathfrak P}_r$) is the ideal of the
ring $\Omega_0$ generated by the elements
$\psi_1,\ldots ,\psi_r$, $\varphi_{r+1},\ldots ,$ $\varphi_s$
(respectively  $\psi_1^p,\ldots,\psi_r^p,\varphi_{r+1}^p,\ldots,
\varphi_s^p$, respectively $\psi_1,\ldots ,\psi_r$).

Let $F\in k[X_1/X_0,\ldots ,X_n/X_0]$. Then
$$
F\in \widetilde{{\mathfrak P}} \quad\mbox{if and only if}\quad
{\cal Z}(F)\supset W\setminus{\cal Z}(X_0).
\eqno (17)
$$

Let (16) holds. Let $i\ge 0$ be an integer.
Put by definition
\begin{eqnarray*}
&&{\cal J}^{(i)}=\{(i_1,\ldots , i_n)\in{\Bbb Z}^n\, :\,
i_1+\ldots + i_s=i\},\\
&&F^{(i)}=\sum_{(i_1,\ldots , i_n)\in{\cal
J}^{(i)}}F_{i_1,\ldots ,i_n}\psi_1^{i_1}\cdot\ldots\cdot \psi_r^{i_r}
\varphi_{r+1}^{i_{r+1}}\cdot\ldots\cdot
\varphi_{s}^{i_{s}}Z_{s+1}^{i_{s+1}}\cdot\ldots\cdot
Z_{n}^{i_{n}},\\
&&\overline{\cal J}=\{(i_1,\ldots , i_n)\in{\Bbb Z}^n\, :\,
i_1=\ldots =
i_r=0\,\&\,i_{r+1},\ldots , i_n\ge 0\},\\
&&\overline{\cal J}^{(i)}=\{(i_1,\ldots , i_n)\in
\overline{\cal J}\, :\,
i_{r+1}+\ldots + i_s=i\},\\
&&\overline{F}=\sum_{(i_1,\ldots , i_n)\in\overline{\cal
J}}F_{i_1,\ldots ,
i_n}\varphi_{r+1}^{i_{r+1}}\cdot\ldots\cdot
\varphi_{s}^{i_{s}}Z_{s+1}^{i_{s+1}}\cdot\ldots\cdot
Z_{n}^{i_{n}}, \\
&&\overline{F}^{(i)}=\sum_{(i_1,\ldots , i_n)\in\overline{\cal
J}^{(i)}}F_{i_1,\ldots ,
i_n}\varphi_{r+1}^{i_{r+1}}\cdot\ldots\cdot
\varphi_{s}^{i_{s}}Z_{s+1}^{i_{s+1}}\cdot\ldots\cdot
Z_{n}^{i_{n}}.
\end{eqnarray*}
Hence $\overline{F}$ is obtained by the substitution of $0$ in place of
$\psi_1,\ldots ,\psi_r$ in $F$, and
$\overline{F}=\sum_{i\ge 0}\overline{F}^{(i)}$, $F=\sum_{i\ge 0}F^{(i)}$.
Further, set
\begin{eqnarray*}
&&\mbox{\rm ord}_{1,n} F=\min\{i_1+\ldots + i_n\,
:\,F_{i_1,\ldots , i_n}\ne
0\},\\
&&\mbox{\rm ord}_{1,s} F=\min\{i\, :\,F^{(i)}\ne
0\},\\
&&\mbox{\rm ord}_{1,s,\Omega} F=\min\{i\,
:\,F^{(i)}\not\in\Omega\},\\
&&\mbox{\rm ord}_{1,s,{\mathfrak q}} F=\min\{i\,
:\,F^{(i)}\not\in{\mathfrak q}\},\\
&&\mbox{\rm ord}_{1,s,\Omega+{\mathfrak q}} F=
\min\{i\,:\,F^{(i)}\not\in\Omega+{\mathfrak q}\},\\
&&\mbox{\rm ord}_{r+1,s} F=\min\{i\, :\,\overline{F}^{(i)}\ne
0\}, \\
&&\mbox{\rm ord}_{r+1,s,{\mathfrak q}} F=\min\{i\,
:\,\overline{F}^{(i)}
\not\in{\mathfrak q}\}.
\end{eqnarray*}
Note here that $\Omega+{\mathfrak q}\subset\Omega_0$ is a linear
subspace.
Hence $\mbox{\rm ord}_{1,n} F$ (respectively $\mbox{\rm
ord}_{1,s} F$,
$\mbox{\rm ord}_{1,s,\Omega}F$,
$\mathop{\rm ord}_{1,s,{\mathfrak q}}F$,
$\mathop{\rm ord}_{1,s,\Omega+{\mathfrak q}}F$,
$\mathop{\rm ord}_{r+1,s} F$,
$\mbox{\rm ord}_{r+1,s,{\mathfrak q}} F$)
is a non--negative integer or $+\infty$.
Obviously
\begin{eqnarray*}
&&\mbox{\rm ord}_{1,n} F\le\mbox{\rm ord}_{1,s} F\le
\mbox{\rm ord}_{1,s,\Omega} F\le
\mbox{\rm ord}_{r+1,s} F\le
\mbox{\rm ord}_{r+1,s,{\mathfrak q}} F, \\
&&\mbox{\rm ord}_{1,s} F\le
\mbox{\rm ord}_{1,s,{\mathfrak q}} F\le
\mbox{\rm ord}_{1,s,\Omega+{\mathfrak q}} F\le
\mbox{\rm ord}_{r+1,s,{\mathfrak q}} F.
\end{eqnarray*}
Besides that,
\setcounter{equation}{17} \begin{eqnarray}
&&\mbox{\rm ord}_{r+1,s,{\mathfrak q}} F\ne+\infty
\quad\mbox{implies}\quad
\mbox{\rm ord}_{r+1,s,{\mathfrak q}} F
\le (s-r)(p-1) \label{18} \\
&&\mbox{\rm ord}_{1,s,{\mathfrak q}} F\ne+\infty
\quad\mbox{implies}\quad
\mbox{\rm ord}_{1,s,{\mathfrak q}} F
\le s(p-1), \label{19} \\
&&\mbox{\rm ord}_{1,s,\Omega+{\mathfrak q}}
F\ne+\infty
\quad\mbox{implies}\quad
\mbox{\rm ord}_{1,s,\Omega+{\mathfrak q}} F
\le s(p-1), \label{20}
\end{eqnarray}

{\it Let $j$ be equal to $\mbox{\rm ord}_{1,s} F$
(respectively
$\mbox{\rm ord}_{1,s,{\mathfrak q}} F$,
$\mbox{\rm ord}_{1,s,\Omega} F$,
$\mbox{\rm ord}_{1,s,\Omega+{\mathfrak q}} F$,
$\mbox{\rm ord}_{r+1,s} F$,
$\mbox{\rm ord}_{r+1,s,{\mathfrak q}} F$). If
$j=+\infty$
then by definition $F^{(j)}=0$ (respectively $F^{(j)}_{{\mathfrak
q}}=0$,
$F^{(j)}_{\Omega}=0$,
$F^{(j)}_{\Omega+{\mathfrak q}}=0$,
$\overline{F}^{(j)}=0$,
$\overline{F}^{(j)}_{{\mathfrak q}}=0$).}
Further, put
\begin{eqnarray*}
&&F^*=F^{(j_0)},\quad\mbox{where}\quad
j_0=\mbox{\rm ord}_{1,s}F,
\\
&&F^*_{{\mathfrak
q}}=F^{(j_1)},\quad\mbox{where}\quad
j_1=\mbox{\rm ord}_{1,s,{\mathfrak q}}F,
\\
&&F^*_{\Omega}=F^{(j_2)},\quad\mbox{where}\quad
j_2=\mbox{\rm ord}_{1,s,\Omega}F,
\\
&&F^*_{\Omega+{\mathfrak q}}=F^{(j_3)},\quad\mbox{where}\quad
j_3=\mbox{\rm ord}_{1,s,\Omega+{\mathfrak q}}F,
\\
&&\overline{F}^*=\overline{F}^{(j_4)},\quad\mbox{where}\quad
j_4=\mbox{\rm ord}_{r+1,s}F,
\\
&&\overline{F}^*_{{\mathfrak q}}
=\overline{F}^{(j_5)},\quad\mbox{where}\quad
j_5=\mbox{\rm ord}_{r+1,s,{\mathfrak q}}F.
\end{eqnarray*}
Let us define also the leading form of an element $F$, see (16),
with respect to
$\psi_1,\ldots ,\psi_r$, $\varphi_{r+1},\ldots,\varphi_s$, $Z_{s+1},\ldots ,
Z_n$
by the formula
$$
\mbox{\rm lf}(F)=\sum_{i_1+\ldots + i_n=\mbox{\rm
ord}_{1,n}F}F_{i_1,\ldots , i_n}
\psi_1^{i_1}\cdot\ldots\cdot \psi_r^{i_r}
\varphi_{r+1}^{i_{r+1}}\cdot\ldots\cdot
\varphi_{s}^{i_{s}}Z_{s+1}^{i_{s+1}}\cdot\ldots\cdot
Z_{n}^{i_{n}}.
$$


Set
$$
{\cal I}=\{(i_1,\ldots , i_n)\in{\Bbb Z}^n\, :\, 0\le i_1,\ldots ,
i_n<p\},
\eqno (21)
$$
and for every $I=(i_1,\ldots , i_n)\in{\cal I}$ set $Z^I=Z_1^{i_1}\cdots
Z_n^{i_n}$.
One can represent an arbitrary element $F\in\overline{k}[[Z_1,\ldots ,Z_n]]$
in the form
$$
F=\sum_{I\in{\cal I}}F_IZ^I,
\eqno (22)
$$
where all $F_I=F_{(i_1,\ldots , i_n)}\in\overline{k}[[Z_1^p,\ldots ,
Z_n^p]]$ are uniquely defined.
If $F\in k[Z_1,\ldots ,$ $Z_n]$ then $F_I\in k[Z_1^p,\ldots ,Z_n^p]$
and $\deg_{Z_1,\ldots ,Z_n}F_I\le\deg_{Z_1,\ldots ,Z_n}F$ for all
$I\in{\cal I}$.




Let us introduce the operators ${\cal D}_j$, $r+1\le j\le s$, of
differentiation
on the ring (15) by the formulas
$$
{\cal D}_j(F)= \det\left(
\begin{array}{cccc}
\partial \psi_1/\partial Z_1 &\ldots & \partial \psi_1/\partial Z_{r} &
\partial \psi_1/\partial Z_{j}\\
\vdots             &\vdots  & \vdots
&\vdots\\
\partial \psi_r/\partial Z_1 &\ldots & \partial \psi_r/\partial Z_{r}
& \partial \psi_r/\partial Z_{j}\\
\partial F/\partial Z_1 &\ldots & \partial F/\partial Z_{r}
&\partial F/\partial Z_{j}
\end{array}
\right).
$$
Obviously if $F\in\widetilde{{\mathfrak P}}$ and $\mbox{\rm
ord}_{r+1,s}F>1$ then ${\cal D}_i(F)\in\widetilde{{\mathfrak
P}}$ for every $r+1\le i\le s$.
We have
$$
{\cal D}_i({\mathfrak q})\subset{\mathfrak q},
\quad
{\cal D}_i({\mathfrak P}_r)\subset{\mathfrak P}_r,\quad
{\cal D}_i(\Omega)=\{0\},\quad r+1\le i\le s.
\eqno (23)
$$
If $F\in \overline{k}[Z_1,\ldots ,Z_n]$ then degree relative to
$Z_1,\ldots ,Z_n$
$$
\deg{\cal D}_i(F)\le \deg F+\deg g_1+\ldots +
\deg
g_r-r-1.
\eqno (24)
$$
If $F\in k[X_1/X_0,\ldots ,X_n/X_0]$  then
${\cal D}_i(F)\in k[X_1/X_0,\ldots ,X_n/X_0]$ for every $r+1\le i\le s$.

\par\medskip\noindent{\bf LEMMA~2}\hspace{0.1em} {\it  Let $F\in{\mathfrak
q}$.
Then, see (22), also $F_I\in{\mathfrak q}$ for every
$I\in{\cal I}$.
}\par\medskip

\noindent{\bf PROOF}\quad Let us represent $F=\sum_{1\le i\le r}a_i\psi_i^p+
\sum_{r+1\le j\le s}b_j\varphi_j^p$ where all  $a_i,b_j$ are from ring
(15). By (22) we have $a_i=\sum_{I\in{\cal I}}a_{i,I}Z^I$,
$b_j=\sum_{I\in{\cal I}}b_{j,I}Z^I$ where all
$a_{i,I},b_{j,I}\in\overline{k}[[Z_1^p,\ldots ,Z_n^p]]$.
Then obviously $F_I=\sum_{1\le i\le r}a_{i,I}\psi_i^p+
\sum_{r+1\le j\le s}b_{j,I}\varphi_j^p$ for every $I\in{\cal I}$.
Therefore, $F_I\in{\mathfrak q}$. The lemma is proved.

\par\medskip\noindent{\bf LEMMA~3}\hspace{0.1em} {\it  Let
$F\in\widetilde{{\mathfrak P}}\cap\overline{k}[Z_1,\ldots
,Z_n]$, and $\nu=\mbox{\rm
ord}_{1,s,{\mathfrak q}}F<\mbox{\rm
ord}_{1,s,\Omega+{\mathfrak q}}F$.
Put
$$
{\cal I}_\nu=\{(i_1,\ldots ,i_r)\in{\Bbb Z}^r\, :\,0\le i_1,\ldots ,
i_r\le p-1\,\&\,
i_1+\ldots + i_r=\nu\}.
$$
Then there exist polynomials $\Delta_{i_1,\ldots
,i_r}\in\overline{k}[Z_1,\ldots ,Z_n]$ for all $(i_1,\ldots
,i_r)\in{\cal I}_\nu$
and a polynomial $\widetilde{F}\in\overline{k}[Z_1,\ldots ,Z_n]$ such that
$$
\Delta^\nu F=\widetilde{F}+\sum_{(i_1,\ldots ,i_r)\in{\cal
I}_\nu}\Delta_{i_1,\ldots ,i_r}\psi_1^{i_1}\cdot\ldots\cdot
\psi_r^{i_r},
\eqno (25)
$$
where for every $(i_1,\ldots ,i_r)\in{\cal I}_\nu$ the degree with
respect to $Z_1,\ldots ,Z_n$
$$
\deg(\Delta_{i_1,\ldots ,i_r}\psi_1^{i_1}\cdot\ldots\cdot\psi_r^{i_r})\le
\deg(F)+\nu(-r+\deg\psi_1+\ldots +\deg\psi_r),
\eqno (26)
$$
and
$$
\mbox{\rm ord}_{1,s,{\mathfrak q}}\widetilde{F}>
\mbox{\rm ord}_{1,s,{\mathfrak q}}F,\quad
\mbox{\rm ord}_{r+1,s}\widetilde{F}=
\mbox{\rm ord}_{r+1,s}F.
\eqno (27)
$$
}\par\medskip

\noindent{\bf PROOF}\quad We have
$$
F^*_{{\mathfrak q}}=\sum_{(i_1,\ldots ,i_r)\in{\cal
I}_\nu}\alpha_{i_1,\ldots ,i_r}\psi_1^{i_1}\cdot\ldots\cdot
\psi_r^{i_r}+Q,
$$
where all $\alpha_{i_1,\ldots ,i_r}\in\overline{k}[[Z_{s+1},\ldots , Z_n]]$
and $Q\in{\mathfrak q}$.
Let us show that for every $(i_1,\ldots ,i_r)\in{\cal I}_\nu$ there
exist a polynomial $\Delta_{i_1,\ldots ,i_r}\in\overline{k}[Z_1,\ldots
,Z_n]$ and
an element $\widetilde{\alpha}_{i_1,\ldots
,i_r}\in\overline{k}[[Z_{s+1},\ldots ,Z_n]]$ such that
\setcounter{equation}{27} \begin{eqnarray}
&&\Delta^\nu\alpha_{i_1,\ldots ,i_r}=\Delta_{i_1,\ldots
,i_r}+\widetilde{\alpha}_{i_1,\ldots ,i_r}, \label{28} \\
&&\mbox{\rm
ord}_{1,s}\widetilde{\alpha}_{i_1,\ldots ,i_r}>0,
\label{29}
\end{eqnarray}
and (26) holds.
Obviously this will imply the assertion of the lemma.

We shall assume without loss of generality that $i_1>0$ and
$\mbox{\rm ord}_{1,s}\alpha_{i_1,\ldots ,i_r}=0$. Now
$$
\frac{\partial F}{\partial Z_j}=\sum_{1\le i\le r}\frac{\partial
F}{\partial\psi_i}
\frac{\partial\psi_i}{\partial Z_j}+Q_j,\quad 1\le j\le r,
$$
where $Q_j\in\overline{k}[[Z_1,\ldots ,Z_n]]$ and $\mbox{\rm
ord}_{1,s,{\mathfrak q}}Q_j\ge\mbox{\rm
ord}_{1,s,{\mathfrak q}}F$.
Hence by Cramer's rule
$$
\Delta\frac{\partial F}{\partial\psi_1}=
\sum_{1\le j\le r}\Delta_{1,j}\frac{\partial F}{\partial Z_j}+Q',\quad
1\le i\le r,
\eqno (30)
$$
where $\Delta_{1,j}\in\overline{k}[Z_1,\ldots ,Z_n]$, $1\le j\le r$;
$Q'\in\overline{k}[[Z_1,\ldots ,Z_n]]$;
$\mbox{\rm ord}_{1,s,{\mathfrak q}}Q'\ge\mbox{\rm
ord}_{1,s,{\mathfrak q}}F$, and
$$
\deg\Bigl(\,\Delta_{1,j}\,\frac{\partial F}{\partial
Z_j}\,\Bigr)\le-r+\deg(F)+\sum_{2\le i\le
r}\deg\psi_i
\eqno (31)
$$
for every $1\le j\le r$.
Set $H=\sum_{1\le j\le r}(\Delta_{1,j}{\partial F}/{\partial Z_j})$.
Now
$$
\nu-1=\mbox{\rm ord}_{1,s,{\mathfrak q}}H<\mbox{\rm
ord}_{1,s,\Omega+{\mathfrak q}}H
$$
by (30) and since
$\mbox{\rm ord}_{1,s}\alpha_{i_1,\ldots , i_n}=0$.
Further,
$$
H^*_{{\mathfrak q}}
=\Delta(x)\sum_{(i_1,\ldots ,i_r)\in{\cal
I}_\nu,\,i_1>0}i_1\alpha_{i_1,\ldots ,i_r}\psi_1^{i_1-1}
\psi_2^{i_2}\cdot\ldots\cdot\psi_r^{i_r}+Q'',
\eqno (32)
$$
where $Q''\in\overline{k}[[Z_1,\ldots ,Z_n]]$ and $\mbox{\rm
ord}_{1,s,{\mathfrak q}}Q''\ge\mbox{\rm
ord}_{1,s,{\mathfrak q}}F$.
If $\nu=1$ then (32) and (31) imply (28) and (29).
If $\nu>1$ we shall use the induction on $\nu$.
Applying the inductive assumption for $H$ we get
$$
\Delta^{\nu-1}(\Delta i_1\alpha_{i_1,\ldots ,i_r})=\Delta'_{i_1-1,i_2,\ldots
, i_r}+\alpha'_{i_1-1,i_2\ldots ,i_r},
$$
where $\mbox{\rm ord}_{1,s}
\alpha'_{i_1-1,i_2\ldots ,i_r}>0$ and by (26)
(with $\nu-1$ in place of $\nu$)
\begin{eqnarray*}
&&\deg(\Delta'_{i_1-1,i_2,\ldots
, i_r}\psi_1^{i_1-1}\psi_2^{i_2}\cdot\ldots\cdot\psi_r^{i_r})\le \\
&&\deg(H)+(\nu-1)(-r+\deg\psi_1+\ldots +\deg\psi_r)\le\\
&&\deg(F)-\deg\psi_1+\nu(-r+\deg\psi_1+\ldots +\deg\psi_r).
\end{eqnarray*}
Put $\Delta_{i_1,\ldots i_r}=i_1^{-1}\Delta'_{i_1-1,i_2,\ldots, i_r}$.
Then (28), (29) hold. The lemma is proved.

\par\medskip\noindent{\bf LEMMA~4}\hspace{0.1em} {\it  Let  $0\ne
F\in\widetilde{{\mathfrak P}}$ and
$$
0<\nu=\mbox{\rm
ord}_{1,s,{\mathfrak q}}F=\mbox{\rm
ord}_{1,s,\Omega+{\mathfrak q}}F<
\mbox{\rm ord}_{r+1,s,{\mathfrak q}}F.
$$
Then there is an integer $1\le j\le r$ such that
$$
\mbox{\rm ord}_{1,s,{\mathfrak q}}
\Bigl(\,\frac{\partial F}{\partial
Z_j}\,\Bigr)
 =\mbox{\rm
ord}_{1,s,\Omega+{\mathfrak q}}
\Bigl(\,\frac{\partial F}{\partial Z_j}\,\Bigr)
=\nu-1,
\eqno (33)
$$
where $\partial/\partial Z_j$ is the differentiation of the ring
$k[[Z_1,\ldots ,Z_n]]=\Omega_0$.
}\par\medskip

\noindent{\bf PROOF}\quad There is $(i_1,\ldots , i_n)\in{\cal I}$,
see (21), such
that $F_{i_1,\ldots , i_n}\ne 0$,
$i_1+\ldots + i_s=\nu$, $i_1+\ldots + i_r<\nu$,
and the sum $i_1+\ldots + i_r=\nu_1$ is maximal
possible. Then $\nu_1>0$ since $\mbox{\rm
ord}_{1,s,\Omega+{\mathfrak q}}F<
\mbox{\rm ord}_{r+1,s,{\mathfrak q}}F$.
For all integers $a\ge b\ge 0$ set
\begin{eqnarray*}
&&J_{a,b}=
\{(i_1,\ldots , i_n)\in{\Bbb Z}^n\, :\, i_1+\ldots +
i_s=a\,\&\,i_1+\ldots +
i_r=b\}, \\
&&Q=\sum_{(i_1,\ldots , i_n)\in J_{\nu,\nu_1}}F_{i_1,\ldots ,
i_n}\psi_1^{i_1}\cdot\ldots\cdot \psi_r^{i_r}
\varphi_{r+1}^{i_{r+1}}\cdot\ldots\cdot
\varphi_{s}^{i_{s}}Z_{s+1}^{i_{s+1}}\cdot\ldots\cdot
Z_{n}^{i_{n}}.
\end{eqnarray*}
Let us represent
$$
\frac{\partial Q}{\partial Z_j}=\sum_{(i_1,\ldots , i_n)\in {\Bbb
Z}^n}Q_{j,i_1,\ldots , i_n}
\psi_1^{i_1}\cdot\ldots\cdot \psi_r^{i_r}
\varphi_{r+1}^{i_{r+1}}\cdot\ldots\cdot
\varphi_{s}^{i_{s}}Z_{s+1}^{i_{s+1}}\cdot\ldots\cdot
Z_{n}^{i_{n}},
$$
where all the coefficients $Q_{j,i_1,\ldots , i_n}\in\overline{k}$.
We have $\mbox{\rm ord}_{1,s}(\partial F^{(i)}/\partial Z_j)\ge
i-1$ for every $i\ge 1$ and $1\le j\le r$, and
$$
\Bigl(\,\frac{\partial}{\partial Z_j}\,\Bigr)({\mathfrak
q})\subset{\mathfrak q}
\eqno (34)
$$
for every $1\le j\le r$.
Hence for every $(i_1,\ldots , i_n)\in J_{\nu-1,\nu_1-1}$ the
coefficient
of $\partial F/\partial Z_j$ at
$\psi_1^{i_1}\cdot\ldots\cdot\psi_r^{i_r}
\varphi_{r+1}^{i_{r+1}}\cdot\ldots\cdot
\varphi_{s}^{i_{s}}Z_{s+1}^{i_{s+1}}\cdot\ldots\cdot
Z_{n}^{i_{n}}$ is $Q_{j,i_1,\ldots , i_n}$.
Therefore, it is
sufficient to prove that
$Q_{j,i_1,\ldots , i_n}\ne 0$ for some $1\le j\le r$ and $(i_1,\ldots ,
i_n)\in J_{\nu-1,\nu_1-1}$.

Recall that $\partial/\partial\psi_i$, $1\le i\le r$, is the natural
differentiation of the ring  $\Omega_0$.
There exists $1\le i\le r$ such that $\partial Q/\partial\psi_i\ne 0$.
Put
\begin{eqnarray*}
&&\mu=\min\{\mbox{\rm ord}_{1,n}({\partial Q}/{\partial
\psi_i})\, :\, 1\le i\le r\}, \\
&&I=\{i\, :\, \mbox{\rm ord}_{1,n}({\partial Q}/{\partial
\psi_i})=\mu\,\&\, 1\le i\le r\}. \\
&&I_j=\{i\, :\, ({\partial\psi_i}/{\partial Z_j})(x)\ne 0\,\&\, 1\le
i\le r\},\quad
1\le j\le r.
\end{eqnarray*}
We have
\setcounter{equation}{34} \begin{eqnarray}
&&\sum_{(i_1,\ldots , i_n)\in J_{\nu-1,\nu_1-1}}Q_{j,i_1,\ldots , i_n}
\psi_1^{i_1}\cdot\ldots\cdot \psi_r^{i_r}
\varphi_{r+1}^{i_{r+1}}\cdot\ldots\cdot
\varphi_{s}^{i_{s}}Z_{s+1}^{i_{s+1}}\cdot\ldots\cdot
Z_{n}^{i_{n}}= \nonumber\\
&&\sum_{i\in I_j}\frac{\partial Q}{\partial
\psi_i}\Bigl(\,\frac{\partial\psi_i}{\partial Z_j}\,\Bigr)^*.\label{35}
\end{eqnarray}
Further,
$$
\mbox{\rm lf}\Bigl(\,\sum_{i\in I_j}\frac{\partial Q}{\partial
\psi_i}
\Bigl(\,\frac{\partial\psi_i}{\partial Z_j}\,\Bigr)^*\,\Bigr)=\sum_{i\in
I}\mbox{\rm
lf}\Bigl(\,
\frac{\partial Q}{\partial \psi_i}\,\Bigr)
\Bigl(\,\frac{\partial\psi_i}{\partial Z_j}\,\Bigr)(x).
\eqno (36)
$$
By (14) there is $1\le j\le r$ such that the sum in the right part of
(36)
is nonzero. Then by (36) the sum in the right part of (35) is nonzero.
The lemma is proved.



\par\medskip\noindent{\bf LEMMA~5}\hspace{0.1em} {\it  Let
$F\in\widetilde{{\mathfrak P}}$ and  $\mbox{\rm
ord}_{r+1,s,{\mathfrak q}}F<+\infty$.
Then there is $r+1\le i\le s$ such that
$$
\mbox{\rm ord}_{r+1,s,{\mathfrak q}}({\cal
D}_i(F))=\mbox{\rm
ord}_{r+1,s,{\mathfrak q}}(F)-1.
\eqno (37)
$$
}\par\medskip

\noindent{\bf PROOF}\quad
%We have ${\cal D}_i({\mathfrak q})\subset
%{\mathfrak q}$ for every $r+1\le i\le s$.
Recall that $\partial/\partial\varphi_j$, $r+1\le j\le s$, is
the natural differentiation of ring (15).
Put $\nu_0=\mbox{\rm ord}_{r+1,s,{\mathfrak q}}F$.
Since $\nu_0<+\infty$ there is $(i_1,\ldots, i_n)\in{\cal
I}\cap{\cal J}^{(\nu_0)}$ such that the coefficient $F_{i_1,\ldots ,
i_n}\ne 0$, see (16).
Therefore, there is $r+1\le j_0\le s$ such that
$\partial \overline{F}^*/\partial\varphi_{j_0}\ne 0$.

By the definition of the operators ${\cal D}_i$, by (14) and
since any linear combination of $\varphi_{r+1},\ldots , \varphi_s$
(with at least one nonzero coefficient) is a local parameter at the point
$x$ the
vectors
$$
({\cal D}_{r+1}(\varphi_i)(x),\ldots , {\cal D}_s(\varphi_i)(x))
\in(\overline{k})^{s-r}, \quad r+1\le i\le s,
$$
are linearly independent over $\overline{k}$.

Set $\nu_1=\min\{\mbox{\rm
ord}_{1,n}(\,\partial\overline{F}^*_{\mathfrak
q}/\partial\varphi_j\,)\, :\,
r+1\le
j\le s\}$ and
$$
J=\{j\, :\,\mbox{\rm
ord}_{1,n}(\,\partial\overline{F}^*_{\mathfrak
q}/\partial\varphi_j\,)=\nu_1\,\&\,r+1\le j\le s\}.
$$
Hence there exists $r+1\le i\le s$ such that
$$
\sum_{j\in J}\mbox{\rm lf}\Bigl(\,
\frac{\partial \overline{F}^*_{\mathfrak q}}{\partial\varphi_j}\,\Bigr)\,
{\cal D}_i(\varphi_j)(x)\ne 0.
$$
Set
$$
J'=\{j\, :\,{\cal D}_i(\varphi_j)(x)\ne 0\,\&\,r+1\le j\le s\}.
$$
But in this case
$$
\sum_{j\in J}\mbox{\rm lf}\Bigl(\,
\frac{\partial \overline{F}^*_{\mathfrak q}}{\partial\varphi_j}\,\Bigr)\,
{\cal D}_i(\varphi_j)(x)=
\mbox{\rm lf}\Bigl(\,\sum_{j\in J'}
\frac{\partial \overline{F}^*_{\mathfrak q}}{\partial\varphi_j}\,
\overline{{\cal D}_i(\varphi_j)}^*\,\Bigr)\ne 0.
$$
Therefore, also $\sum_{j\in J'}(\partial
\overline{F}^*_{\mathfrak q}/\partial\varphi_j)\,\overline{{\cal
D}_i(\varphi_j)}^*\ne 0$.
This implies
$$
\overline{({\cal D}_i(F))}^*_{\mathfrak q}=\sum_{j\in J'}
\frac{\partial \overline{F}^*_{\mathfrak q}}
{\partial\varphi_j}\,\overline{{\cal
D}_i(\varphi_j)}^*.
$$
Hence (37) holds. The lemma is proved.




















\medskip Let us define $\delta$ to be the least integer such that
$$
\delta\ge\frac{\deg g_1\cdot\ldots\cdot \deg g_r\deg
f_{r+1}\cdot\ldots\cdot \deg f_{s}}{\deg W}.
$$
The following lemma is similar to Lemma~18 Section~10 \cite{2}.
Still due to its importance we repeat the proof.

\par\medskip\noindent{\bf LEMMA~6}\hspace{0.1em} {\it  The inequality
$\mbox{\rm ord}_{r+1,s}\rho_{r+1}\le
\delta$ holds.
}\par\medskip

\noindent{\bf PROOF}\quad
Suppose contrary. Denote by $\psi_{i,1}$, $1\le i\le r$, $\varphi_{j,1}$,
$r+1\le j\le s$
the forms of the least degree
(or which is the same the non--zero homogeneous in $Z_1,\ldots ,Z_n$
components of the least degree)
in the ring $\overline{k}[Z_1,\ldots ,Z_n]$ of the polynomials
$\psi_{i}$, $1\le i\le r$, $\varphi_{j}$, $r+1\le j\le s$ respectively.

By Lemma~16 \cite{2} (it is valid in arbitrary characteristic) and
(12), (13)
for every $r+1\le j\le s$
there are polynomials $q$, $h_1,\ldots , h_s\in \overline{k}[Z_1,\ldots
,Z_n]$
such that
$$
h_1\psi_1+\ldots + h_r\psi_r+h_{r+1}\rho_{r+1}+\ldots + h_{s}\rho_{s}
=q\varphi_j^\delta
$$
and the polynomial $q$ is not vanishing on $W\setminus{\cal Z}(X_0)$.
Replacing if necessary the point $x$ by a point $x_1\in W$ from a
sufficiently small neighborhood in the Zariski topology
of the point $x$ (it is reduced to a linear automorphism $X_i\mapsto
X_i+\lambda_iX_0$,
$\lambda_i\in \overline{k}$, $1\le i\le n$, of ${\Bbb P}^n(\overline{k})$)
we can assume without loss of generality that $q(x)\ne 0$.
So $q$ is an invertible element of the ring of formal power series
(15).
Therefore there exist polynomials $\widetilde{h_i}\in
\overline{k}[Z_1,\ldots ,Z_n]$
which are approximations of the elements
$h_i/q$ from ring (15) such that
$$
\widetilde{h}_1\psi_1+\ldots + \widetilde{h}_r\psi_r+
\widetilde{h}_{r+1}\rho_{r+1}+\ldots + \widetilde{h}_{s}\rho_{s} =\Phi_j,
$$
where $\Phi_j\in \overline{k}[Z_1,\ldots ,Z_n]$ is a polynomial with the
form of the least degree in the ring (15)
$\varphi_j^\delta$. Hence the form of the least degree of the polynomial
$\Phi_j$ in the ring $\overline{k}[Z_1,\ldots ,Z_n]$
is $\varphi_{j,1}^\delta$.

By our assumption and (17) applied to $F=\rho_{r+1}$ all the homogeneous
components of degree
less than $\delta+1$ of the product $\widetilde{h}_{r+1}\rho_{r+1}$ belong
to the ideal ${\mathfrak P}_r$, see above.
Therefore we can assume without loss of generality that
$\widetilde{h}_{r+1}=0$.

Hence the linear forms $\psi_{i,1}$, $1\le i\le r$, $\varphi_{j,1}$, $r+1\le
j\le s$
belong to the radical of the ideal generated by the forms of the least
degree of the elements from the
ideal
$$
(\psi_{1},\ldots , \psi_{r},\rho_{r+2},\ldots , \rho_{s}
)\subset \overline{k}[Z_1,\ldots ,Z_n].
$$
But by definition this radical
is the ideal of the tangent cone of the algebraic
variety
$W={\cal Z}(\psi_{1},\ldots , \psi_{r},\rho_{r+2},\ldots ,
\rho_{s} )$ at the point $x$.
Denote this tangent cone by $\mbox{\rm con}(x,W)$. It is known
that $\dim \mbox{\rm con}(x, W)=
\dim_x W$ where $\dim_x W$ is the maximum of dimensions of irreducible
components of $W$ containing
the point $x$ (the direct proof of this fact can be found in \cite{5};
although the case of zero--characteristic is considered in \cite{5} the last
proof is valid in arbitrary characteristic).

Now we have $\dim_x W=n-s+1$ by conditions (a)--(d). On the other hand,
there are $s$ linearly independent linear forms
from the ideal of $\mbox{\rm con}(x,W)$. Hence $\dim\mbox{\rm
con}(x, W)\le n-s<\dim_x W$.
This contradiction proves the lemma.


\section{Description of the main recursion.}\label{s2}


Set $\rho=\rho_{r+1}$.
Now we are ready to describe a new recursion on $\rho$. At the end of this
recursion
using the obtained element $\rho$ we shall construct $g_{r+1}$ and
homogeneous
polynomials $f_{r+1,r+2},\ldots , f_{r+1,s}$, see Section~1.
The base of the recursion is $\rho=\rho_{r+1}$.
The recursion starts from the first step.
At the beginning of each step $\mbox{\rm ord}_{1,s}\rho>1$
and $\mbox{\rm ord}_{r+1,s}(\rho)<+\infty$.

Let us describe the general step of the recursion.
Obviously one and only one of the following four cases holds:
\begin{enumerate} \renewcommand{\labelenumi}{\arabic{enumi})}
\item $\mbox{\rm ord}_{1,s,{\mathfrak q}}(\rho)=+\infty$,

\item $\mbox{\rm ord}_{1,s,{\mathfrak q}}(\rho)<\mbox{\rm
ord}_{1,s,\Omega+{\mathfrak q}}(\rho)\le\mbox{\rm
ord}_{r+1,s,{\mathfrak q}}(\rho)=+\infty$,

\item $\mbox{\rm ord}_{1,s,{\mathfrak q}}(\rho)=\mbox{\rm
ord}_{1,s,\Omega+{\mathfrak q}}(\rho)<\mbox{\rm
ord}_{r+1,s,{\mathfrak q}}(\rho)=+\infty$,

\item $\mbox{\rm ord}_{r+1,s,{\mathfrak q}}(\rho)<+\infty$.
\end{enumerate}
{\it If 1) (respectively 2), 3), 4))
holds at the beginning of the recursion
step then we shall say for brevity that this step is of
type 1) (respectively 2), 3), 4)).}

Assume that 1) holds. Then $\rho\in{\mathfrak q}$. Let us represent
$\rho=\sum_{I\in{\cal I}}\rho_IZ^I$, where all
$\rho_I\in\overline{k}[Z_1^p,\ldots ,Z_n^p]$.
Then by Lemma~2 all $\rho_I\in{\mathfrak q}$. Hence
$\rho_I^{1/p}\in\overline{k}[Z_1,\ldots ,Z_n]\cap\widetilde{{\mathfrak P}}$
since $\widetilde{{\mathfrak P}}$ is a prime ideal.
There is $I_0\in{\cal I}$ such that $\mbox{\rm
ord}_{r+1,s}\rho_{I_0}\le\mbox{\rm ord}_{r+1,s}\rho$.
Put $\rho'=\rho_{I_0}^{1/p}$. Now
$$
0<\mbox{\rm
ord}_{r+1,s}(\rho')\le\Bigl(\,\frac{1}{p}\,\Bigr)\mbox{\rm
ord}_{r+1,s}\rho,\quad \deg\rho'\le
\Bigl(\,\frac{1}{p}\,\Bigr)\deg\rho.
\eqno (38)
$$
If $\mbox{\rm ord}_{r+1,s}(\rho')=1$ this step is final.
We replace $\rho$ by $\rho'$. If $\mbox{\rm ord}_{r+1,s}(\rho')>1$
we proceed to the next step of the recursion.

Assume that 2) holds. Then we put $F=\rho$ in the statement of Lemma~3.
Applying this lemma we get a polynomial $\widetilde{F}$. Set
$\rho'=\widetilde{F}$.
Then
$$
\mbox{\rm ord}_{1,s,{\mathfrak q}}\rho'>\mbox{\rm
ord}_{1,s,{\mathfrak q}}\rho,\quad\mbox{\rm
ord}_{r+1,s}\rho'=\mbox{\rm ord}_{r+1,s}\rho
\eqno (39)
$$
and by (25) and (26) the inequality
$$
\deg\rho'\le\deg(F)+
\mbox{\rm ord}_{1,s,{\mathfrak q}}(\rho)(-r+\deg\psi_1+\ldots
+\deg\psi_r)
\eqno (40)
$$
holds.
We replace $\rho$ by $\rho'$ and proceed to the next step of the recursion.

Assume that 3) holds. Then we put $F=\rho$ in the statement of Lemma~4.
Applying this lemma we get an index $1\le j\le r$ such that (33) holds.
Set $\rho'=\partial F/\partial Z_j$. Then by (33)
\setcounter{equation}{40} \begin{eqnarray}
&&\deg\rho'\le-1+\deg\rho,\quad\mbox{\rm ord}_{1,s,{\mathfrak
q}}\rho'=
-1+\mbox{\rm ord}_{1,s,{\mathfrak q}}\rho, \label{41} \\
&&\mbox{\rm ord}_{1,s,{\mathfrak q}}(\rho')=\mbox{\rm
ord}_{1,s,\Omega+{\mathfrak q}}(\rho'). \label{42}
\end{eqnarray}
If $\mbox{\rm ord}_{r+1,s}(\rho')=1$ this step is final.
We replace $\rho$ by $\rho'$. If $\mbox{\rm ord}_{r+1,s}(\rho')>1$
we proceed to the next step of the recursion.
Notice that now at the next step 3) or 4) holds.

Assume that 4) holds. Then we put $F=\rho$ in the statement of Lemma~5.
Applying this lemma we get an index $1\le i\le r$ such that
(37) holds.
Set $\rho'={\cal D}_i(F)$. Then by (33)
\setcounter{equation}{42} \begin{eqnarray}
&&\mbox{\rm ord}_{r+1,s,{\mathfrak q}}(\rho')=\mbox{\rm
ord}_{r+1,s,{\mathfrak q}}(\rho')-1, \label{43} \\
&&\deg(\rho')\le\deg(F)+\deg(\psi_1)+\ldots +\deg(\psi_r)-r-1. \label{44}
\end{eqnarray}
If $\mbox{\rm ord}_{r+1,s}(\rho')=1$ this step is final.
We replace $\rho$ by $\rho'$.
If $\mbox{\rm ord}_{r+1,s}(\rho')>1$
we proceed to the next step of the recursion.
Notice that now at the next step again 4) holds.

The recursion is described completely.  In the next lemma we show that it
terminates after a finite number of steps.  Notice that for every step of
the
recursion if $\rho\in k[X_1/X_0,\ldots , X_n/X_0]$ at the beginning of the
step then obviously $\rho'\in k[X_1/X_0,\ldots , X_n/X_0]$ at the end of
this
step (for the step of type 1) this follows from the fact that the field
$k$ is
perfect). We have $\rho_{r+1}\in k[X_1/X_0,\ldots , X_n/X_0]$. Hence at the
end of the recursion the element $\rho\in k[X_1/X_0,\ldots , X_n/X_0]$.

\par\medskip\noindent{\bf REMARK~2}\hspace{0.1em} {\it  The analog of
condition 4) for zero--characteristic is $\mbox{\rm
ord}_{r+1,s}\rho<+\infty$.
Only this case is used in the construction of \cite{2} Section~10.
Thus, in comparison with zero--characteristic we need to
consider three more cases in the recursion.
}\par\medskip



\par\medskip\noindent{\bf LEMMA~7}\hspace{0.1em} {\it  The described
recursion terminates after a finite number of steps.
At the end of the recursion $\mbox{\rm
ord}_{1,s}(\rho)=\mbox{\rm ord}_{r+1,s}(\rho)=1$, and
if $\delta<p$ then
$$
\deg\rho\le\deg(\rho_{r+1})+(\delta-1)(-r-1+\deg\psi_1+\ldots +\deg\psi_r),
\eqno (45)
$$
and for an arbitrary $\delta$
$$
\deg\rho\le\frac{\deg(\rho_{r+1})}{p^{\mu_1}}+
\Bigl(\,(s-r)(p-1)+\frac{1-p^{1-\mu}}{1-p^{-1}}\lambda\,\Bigr)
(-r+\deg\psi_1+\ldots +\deg\psi_r),
\eqno (46)
$$
where $\mu,\mu_1$ are integers such that
\setcounter{equation}{46} \begin{eqnarray}
&&1\le\mu\le 1+(\log\delta/\log p), \label{47}\\
&&\mu_1=\max\{\mu-2,0\},\label{48}
\end{eqnarray}
and
$$
\lambda=1+2+\ldots + s(p-1)=s(p-1)(s(p-1)+1)/2\le s^2p^2/2.
$$
}\par\medskip

\noindent{\bf PROOF}\quad Let us show that there exist an integer $\mu\ge
1$, two finite  sequences  of integers
$a_{2,i},a_{1,i}$, $1\le i\le\mu-1$ (these sequences are empty if $\mu=1$),
and integers $a_{3,\mu},a_{4,\mu+1}$
satisfying the following properties:
\begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})}
\item $a_{2,i}>0$, for all $2\le i\le\mu-1$, and if $\mu\ge 2$ then
$a_{2,1}\ge 0$,
\item $a_{1,i}>0$ for all $1\le i\le\mu-2$, and if $\mu\ge 2$ then
$a_{1,\mu-1}\ge 0$,
\item $a_{3,\mu}\ge 0$,
\item $a_{4,\mu+1}\ge 0$,
\item in the described recursion step $j$ (recall that $j\ge 1$) is of type
2) if and only if there is $0\le\nu\le\mu-2$ such
that
$$
\sum_{1\le i\le\nu}(a_{2,i}+a_{1,i})<j\le a_{2,\nu+1}+
\sum_{1\le i\le\nu}(a_{2,i}+a_{1,i}),
$$
\item in the described recursion step $j$ is of type
1) if and only if there is $0\le\nu\le\mu-2$ such that
$$
 a_{2,\nu+1}+\sum_{1\le i\le\nu}(a_{2,i}+a_{1,i})<j\le
\sum_{1\le i\le\nu+1}(a_{2,i}+a_{1,i}),
$$
\item in the described recursion
step $j$ is of type 3) if and only if
$$
\sum_{1\le i\le\mu-1}(a_{2,i}+a_{1,i})<j\le
a_{3,\mu}+\sum_{1\le i\le\mu-1}(a_{2,i}+a_{1,i}),
$$
\item finally, in the described recursion
step $j$ is of type 4) if and only if
$$
a_{3,\mu}+\sum_{1\le i\le\mu-1}(a_{2,i}+a_{1,i})<j\le
a_{4,\mu+1}+a_{3,\mu}+\sum_{1\le i\le\mu-1}(a_{2,i}+a_{1,i}).
$$
\end{enumerate}
Indeed, assume that a step of type 3)
(respectively 4)) is not final. Then, as we have noticed above,
the following step is of type 3) or 4) (respectively 4)).
Hence by (20) and (41) (respectively (18) and (43)) there might
be only a finite number of steps of type 3) (respectively 4)).
So now, saying less formally, we need to prove that
in the described recursion after a nonnegative finite number of iterations
with steps of type 1) and 2) we shall come to a step of type
3) or 4), or to the end of the recursion.

By (19) and (39)
there might be only a finite number of subsequent steps
of type 2).
After any step of type 2) (respectively 1)) we have
$\mbox{\rm ord}_{r+1,s}(\rho')\le\mbox{\rm
ord}_{r+1,s}(\rho)$ (respectively
$\mbox{\rm ord}_{r+1,s}(\rho')\le p^{-1}\mbox{\rm
ord}_{r+1,s}(\rho)$) by (39) (respectively (23)).
The integer $\mbox{\rm ord}_{r+1,s}(\rho)\ge 1$ at the beginning
of each step.
Hence in the described recursion there might be only a
finite number of subsequent steps of types 2) or 1).
Thus, the existence of an integer $\mu\ge
1$, two finite sequences of integers
$a_{2,i},a_{1,i}$, $1\le i\le\mu-1$, and integers $a_{3,\mu},a_{4,\mu+1}$
satisfying properties (i)--(viii) is proved.
By (38)--(44) if $\delta<p$ then only the steps of type 4) occur in
the recursion, i.e., in this case $\mu=1$ and $a_{3,\mu}=0$.


Let $\delta$ be arbitrary.
Recall that $\mbox{\rm ord}_{r+1,s}\rho_{r+1}\le\delta$ by
Lemma~6.
Therefore, by (39) and (23) the inequality $\delta/p^{\mu-1}\ge 1$
holds.
Hence (47) is satisfied.
Further, by (19) and (39) we have $a_{2,i}\le s(p-1)$
for every $1\le i\le\mu-1$. Recall that $a_{1,i}\ge 1$ for $1\le i\le\mu-2$.
Hence by (38), (39), (40)
after the first $\sum_{1\le i\le\mu-1}(a_{2,i}+a_{1,i})$
steps of the recursion for the obtained element $\rho$ we have the estimation
$$
\deg\rho\le\frac{\deg(\rho_{r+1})}{p^{\mu_1}}+
\Bigl(\,\frac{1-p^{1-\mu}}{1-p^{-1}}\,\Bigr)\lambda
(-r+\deg\psi_1+\ldots +\deg\psi_r).
\eqno (49)
$$
By (21) the same bound (49) is satisfied after the first
$a_{3,\mu}+\sum_{1\le  i\le\mu-1}(a_{2,i}+a_{1,i})$ steps.
By (18) we have $a_{4,\mu+1}\le(s-r)p$.
Finally, by (43), (44) and (49) at the end of the recursion
estimation (46) holds.

Assume that $\delta<p$. Then the recursion consists only of steps of type 4),
see above.
Hence by (43), (44) at the end of the recursion
(45) is fulfilled. The lemma is proved.



\medskip Let $\rho\in\overline{k}[Z_1,\ldots ,Z_n]$ be the polynomial
obtained at the end of the described recursion.
Now our aim is to construct $g_{r+1}$ and polynomials $f_{r+1,r+2},\ldots,
f_{r+1,s}$,
i.e., to describe the recursion from Section~1.
As we have seen $\rho\in
k[X_1/X_0,\ldots ,$ $X_n/X_0]\cap\widetilde{{\mathfrak P}}$.  Hence by
(17)
the element $\rho$ vanishes on $W\setminus{\cal Z}(X_0)$.  We have
$\mbox{\rm ord}_{r+1,s}\rho=1$.
Therefore, by Lemma~5 there is $1\le i\le r$ such that
$$
\mbox{\rm
ord}_{r+1,s}{\cal D}_i(\rho)=0.
\eqno (50)
$$
Now (17) implies
that ${\cal D}_i(\rho)$ does not vanish on $W\setminus{\cal Z}(X_0)$.
Therefore, by (50) there is
a non--empty open  in the Zariski topology
subset $S\subset W$ such that every point $y\in S$ is
a smooth point of $W$ and
the differentials at the point $y$
$$
d_y\psi_1,\ldots , d_y\psi_r,d_y\rho
$$
are linearly independent over $\overline{k}$.

Set $d_{r+1}=
\deg(\rho)$ and
$$
g_{r+1}=\rho X_0^{d_{r+1}}.
\eqno (51)
$$
Then $g_{r+1}\in k[X_0,\ldots ,X_n]$ is a homogeneous polynomial.
Further, for every $y\in S$ the point $y$ is smooth on the
algebraic variety ${\cal Z}(g_1,\ldots , g_{r+1})$ and $\dim_y{\cal
Z}(g_1,\ldots , g_{r+1})=n-r-1$.

Recall the definition from \cite{2}, Section~2.
\par\medskip\noindent{\bf DEFINITION~4}\hspace{0.1em} {\it
Let $0<j\le s$ be an integer. Consider a sequence of
homogeneous polynomials
$\theta_i\in {\mathfrak P}$, $1\le i\le j$.
Let $0\le u<j$ be an integer. Denote by
${\cal W}={\cal W}({\mathfrak P},u)$ the set of all
irreducible components $E$ defined over $k$ of the algebraic variety
${\cal Z}(\theta_1,\ldots , \theta_u)$ such that $E\supset W$.
Suppose that
for every integer $0\le u<j$
the polynomial $\theta_{u+1}\in {\mathfrak P}$
is  such that for every irreducible
component $E\in {\cal W}$ the
algebraic variety
$$
E\not\subset {\cal Z}(\theta_{u+1})
$$
and if $u\ge 1$ then $\deg \theta_{u+1}\ge \deg \theta_u$.
Then any such sequence $\theta_1,\ldots , \theta_j$ will be called weak
${\mathfrak P}$--sequence (of length $j$).
}\par\medskip

Note that every ${\mathfrak P}$--sequence is a weak
${\mathfrak P}$--sequence.


There are two weak ${\mathfrak P}$--sequences $\theta_{1,1},\ldots ,
\theta_{1,s}$ and
$\theta_{2,1},\ldots , \theta_{2,s}$ such that
\begin{itemize}
\item the polynomials $\theta_{1,1},\ldots , \theta_{1,r+1}$ coincides  with
$g_1,\ldots , g_{r+1}$ up to
a permutation,
\item the polynomials $\theta_{2,i}=\eta f_i$, $1\le i\le s$,
for a homogeneous polynomial $\eta$ which is not vanishing on $W$,
\item $\deg \theta_{1,i}\le \deg \eta$ for all $1\le i\le r+1$,
\item $\deg \theta_{1,i}>\deg \theta_{2,s}$ for all $r+2\le i\le s$.
\end{itemize}
Let us apply Lemma~1 \cite{2} to these ${\mathfrak P}$-sequences.
We get the third ${\mathfrak P}$-sequence
$\theta_{3,1},\ldots ,$ $\theta_{3,s}$.
Recall that by Lemma~1 \cite{2}
for every $1\le j\le s$ the element  $\theta_{3,j}$ belongs to the union
of the ideals $(\theta_{1,1},\ldots,
\theta_{1,s})\cup(\theta_{2,1},\ldots,\theta_{2,s})\subset k[X_0,\ldots ,
X_n]$, and
the degree $\deg\theta_{3,j}=\min\{\deg\theta_{1,j},\deg\theta_{2,j}\}$.
Hence $\theta_{3,j}\in(\theta_{2,1},\ldots,\theta_{2,s})$, and
$\eta$ divides $\theta_{3,j}$ for every $r+1\le j\le s$.

Set $f_{r+1,j}=\theta_{3,j}/\eta$,
$r+2\le j\le s$. Let us
replace in conditions (a)--(d) $r$ by $r+1$ and the point $x$ by a point
$y$.
Then by Lemma~1 \cite{2} there is a point $y\in S$ such that (a)--(d)
are satisfied for the constructed homogeneous polynomials
$g_1,\ldots , g_{r+1}, f_{r+1,r+2},\ldots , f_{r+1,s}$ at the point
$y$. This finishes the description of the
recursion step and all the recursion from Section~1.

Thus, finally we shall construct a system of local parameters $\psi_1,\ldots
,\psi_s$ of the algebraic variety $W$ in some
point $x\in W$.

\par\medskip\noindent{\bf LEMMA~8}\hspace{0.1em} {\it  There is a constant
$0\le C\in {\Bbb R}$ and there are $e_j\in
{\Bbb
R}$, $1\le j\le s$
such that the inequalities
\setcounter{equation}{51} \begin{eqnarray}
&&\deg g_j\le e_j\deg f_j,\quad 1\le j\le s,  \label{52} \\
&& e_j\le n^{2^{s^C}}, \quad 1\le j\le s  \label{53}
\end{eqnarray}
hold
}\par\medskip

\noindent{\bf PROOF}\quad
Suppose that inequalities (52) hold for all $1\le j\le r$
where $0\le r<s$. Then
by Theorem~1 from \cite{2} and Lemma~6 we have
$$
\mbox{\rm ord}_{r+1,s}(\rho_{r+1})\le \delta\le
1+e_1\cdot\ldots\cdot e_r\delta_1\cdot\ldots\cdot
\delta_s/\deg W\le
1+a_se_1\cdot\ldots\cdot e_r,
$$
where
$$
a_s<n^{2^{s^{C_1}}}
\eqno (54)
$$
for an absolute constant $0<C_1\in{\Bbb R}$,
see Theorem~1 from \cite{2}.
The inequalities (45) for $\delta<p$ and (46) for $\delta\ge p$ imply
$$
\deg\rho\le\deg(\rho_{r+1})+(n\delta+1)^2(-r+\deg\psi_1+\ldots +\deg\psi_r)
\eqno (55)
$$
(for both cases $\delta<p$ and
$\delta\ge p$).
Recall that $\deg\rho_{r+1}=\delta_{r+1}$.
Hence from (55) we get
$$
\deg g_{r+1}\le\delta_{r+1}+(n(1+a_se_1\cdot\ldots\cdot e_r)+1)^2\Bigl(\,
-r+\sum_{1\le j\le r}e_j\delta_j\,\Bigr).
$$
Put
$$
e_{r+1}=1+(n(1+a_se_1\cdot\ldots\cdot e_r)+1)^2(e_1+\ldots + e_r).
$$
Then (52) holds for $j=r+1$.
Now the existence of the constant $C$ such that inequalities (52) and
(53) hold for all $1\le j\le s$
follows from  (54)  (the cases when $n=1$ or $s=1$ are trivial).
The lemma and Theorem~1 are proved.



\newpage

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\end{document}