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\begin{document}

 \title{Double-Exponential Lower Bound for
the Degree of Any System of Generators of a
        Polynomial Prime Ideal}
\author{Alexander L.~Chistov%
 \\[2ex]
St.~Petersburg Department of Steklov Mathematical Institute\\
of the Academy of Sciences of Russia\\
Fontanka 27, St.~Petersburg 191023, Russia,\\
e-mail: alch@pdmi.ras.ru }
\date{\Large 2008}

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

\maketitle

\begin{abstract}
Consider a polynomial ring $A$ in $n+1$ variables over an arbitrary
infinite field $k$. We prove that for all sufficiently big $n$ and
$d$ there is a homogeneous prime ideal ${\mathfrak p}\subset A$
satisfying the following conditions.  The ideal ${\mathfrak p}$
corresponds to a defined over $k$ and irreducible over
$\overline{k}$ component of a projective algebraic variety given by
a system of homogeneous polynomial equations with polynomials from
$A$ of degrees less than $d$. Any system of generators of ${\mathfrak
p}$ contains a polynomial of degree at least $d^{2^{cn}}$ for an
absolute constant $c>0$ which can be computed efficiently.
This solves an important old problem
in effective algebraic geometry. For the
case of finite fields we obtain a slightly less strong result.
\end{abstract}




\newpage
\section*{Introduction}

In the classical paper  \cite{7} D. Hilbert proved that any polynomial ideal
is
generated by a finite number of polynomials. Hence it is true for
a prime polynomial ideal ${\mathfrak p}$ of an irreducible component of an
algebraic variety given
by a system of polynomial equations in $n$ unknowns of degrees less than $d$.
The problem arose to give upper and lower bounds for degrees
of the generators of ${\mathfrak p}$ as functions of $d$ and $n$.
Notice that the degree of the ideal ${\mathfrak p}$ itself
(see the definition of the degree of an ideal below) is bounded from
above by $d^n$ by the B\'ezout theorem.
It is known that there is a system of generators of ${\mathfrak p}$ with
degrees $d^{2^{O(n)}}$, see Remark~2 below
(the case of nonhomogeneous prime ideal is easily reduced to the one of the
homogeneous prime ideal).
One of the most important open problem in effective algebraic geometry
has been to obtain the similar lower bound
for degrees of any system of generators
of an ideal ${\mathfrak p}$.
In this paper we solve this old
problem over an arbitrary infinite field,
see Theorem~1 below.
This implies also the same lower bound for the stabilization
of the characteristic function of a homogeneous ${\mathfrak p}$, see
Theorem~1
and Remark~1.

So far double--exponential lower bounds have been known for binomial
ideals, see Section~1 for more detail. These ideals are far from being
prime: they have many primary components. All the attempts to reduce
the case of a prime ideal to the known one of binomial ideals
failed. Our construction is simple but ingenious in this sense.
We managed to fulfill this reduction! The key to all is Lemma~2.
But how did we make this discovery? Not occasionally. We
investigated the problem of effective normalization of algebraic
varieties. It turns out that the construction of the normalization
of an algebraic variety is reduced to solving a linear equation
$aX+bY+cZ=0$ over a polynomial ring. Simultaneously as one of
the consequence of our results we got Lemma~2. We hope to return to
the normalization of an algebraic variety from algorithmic point of
view in one of the next papers. Notice that the problem of solving
the equality $aX+bY+cZ=0$ over a polynomial ring is close to
the one formulated at the end of the Introduction.

Let $k$ be a field with algebraic closure $\overline{k}$. Let $n\geqslant
0$ and
$X_0,\ldots , X_n$ be variables.
Let ${\mathfrak a}\subset k[X_0,\ldots , X_n]=A$ be a homogeneous ideal
Denote by $h({\mathfrak p},m)=\dim_k(A/{\mathfrak a})_m$ the
characteristic function of the ideal ${\mathfrak a}$,
where $(A/{\mathfrak a})_m$ is the vector space of the homogeneous elements
of degree $m\geqslant 0$ of the ring
$A/{\mathfrak a}$.
Recall that there is a polynomial $P=
\sum_{0\leqslant i\leqslant n-s}p_iZ^i$, all $p_i\in{\Bbb Q}$,
of degree $\deg P=n-s$ such that
$h({\mathfrak p},m)=p(m)$ for all sufficiently big $m>0$.
By definition  $P$ is the Hilbert polynomial $A/{\mathfrak a}$.
The degree of the ideal ${\mathfrak a}$ can be defined by the formula
$\deg{\mathfrak a}=(n-s)!p_{n-s}$.
Denote by ${\cal Z}({\mathfrak a})\subset{\Bbb P}^n(\overline{k})$
the set of all common zeroes of the
polynomials from ${\mathfrak a}$ in ${\Bbb P}^n(\overline{k})$ (the
similar notations will be used below with other
ideals or family of polynomials and other projective or affine spaces).

Below in the statements of the theorem, the proposition, lemmas
and the corollary the
existence of constants
$c,n_0,d_0$,  $c_0$,  $c_1$
and so on is claimed. All these constants can be
computed efficiently. We prove the following result.

\par\medskip\noindent{\bf THEOREM~1}\hspace{0.1em} {\it  There are constants
$c>0$, $n_0>0$, $d_0>0$ such that for every
infinite field $k$ for all integers
$n>n_0$ and $d>d_0$ there are homogeneous polynomials $f_1,\ldots ,f_\nu\in
A$,  $\nu\leqslant n$, with all
degrees $\deg f_i<d$, $1\leqslant i\leqslant \nu$, and a
homogeneous prime ideal ${\mathfrak p}\subset A$
satisfying the following conditions.
\begin{enumerate} \renewcommand{\labelenumi}{(\alph{enumi})}
\item The set of zeroes ${\cal Z}({\mathfrak p})\subset{\Bbb
P}^n(\overline{k})$ of the ideal ${\mathfrak p}$
coincides with a defined over $k$ and
irreducible over $\overline{k}$ component of the
projective algebraic variety ${\cal Z}(f_1,\ldots ,f_\nu)\subset{\Bbb
P}^n(\overline{k})$ of all common zeroes of the polynomials $f_1,\ldots
,f_\nu$. Hence
the ideal $\overline{k}\otimes_k{\mathfrak
p}\subset\overline{k}\otimes_kA$ is prime.
Moreover, ${\mathfrak p}$ is a primary component of the
ideal $(f_1,\ldots ,f_\nu)\subset A$,
the height $\mbox{\rm ht}({\mathfrak p})=\nu$ and
the projective algebraic variety
${\cal Z}(f_1,\ldots , f_\nu)$ has exactly two irreducible over
$\overline{k}$ components: ${\cal Z}({\mathfrak p})$ and some
linear subspace ${\cal L}\subset{\Bbb P}^n(\overline{k})$
defined over $k$.
\item  Suppose that the characteristic function $h({\mathfrak p},m)$ is
stable for
$m\geqslant m_0$, i.e., coincides with the
Hilbert polynomial of $A/{\mathfrak p}$ for these $m$. Then $m_0\geqslant
d^{2^{cn}}$.
\item Let  $a_1,\ldots , a_m$ be an arbitrary system of generators
of the ideal ${\mathfrak p}$. Then
the maximal degree $\max_{1\leqslant
i\leqslant
m}\deg_{X_0,\ldots , X_n}a_i\geqslant d^{2^{cn}}$.
\end{enumerate}
}\par\medskip


\medskip It is quite probable that one can remove
the word ``infinite'' in the statement of the theorem
and obtain the result for an arbitrary field, see Remark~5 at the end
of Section~4.
Also here Remark~4 at the beginning of Section~3 may be useful.
But in this paper we prove only the following result.



\par\medskip\noindent{\bf PROPOSITION~1}\hspace{0.1em} {\it  Let us replace
in the statement of Theorem~1 the words
``infinite field''
by ``finite field''
and $d^{2^{cn}}$ by $d^{2^{c\sqrt{n}}}$ in conditions (b) and (c).
Then this new statement of theorem holds for finite fields.
}\par\medskip








\par\medskip\noindent{\bf REMARK~1}\hspace{0.1em} {\it  It is known, see
\cite{5} Sections~6~and~7,
that for any homogeneous ideal
${\mathfrak a}\subset A$ the following assertion is satisfied.
If the characteristic function $h({\mathfrak a},m)$ is stable for
$m\geqslant m_0$ where $m_0\geqslant 0$ then for any admissible
ordering of monomials the reduced Gr\"obner basis of ${\mathfrak a}$ with
respect to
this ordering of monomials
consists of homogeneous polynomials of degrees at most $m_0+1$.
Hence there is a system of generators of ${\mathfrak a}$ with degrees at most
$m_0+1$.
}\par\medskip


Here probably one needs to give some details related  to the last remark.
We shall use the notation from \cite{5} (but replace $n$ by $n+1$).
In  \cite{5} the constant $b_0\geqslant b_1\geqslant\ldots\geqslant
b_{n+2}=0$
corresponding  to the ideal  $I={\mathfrak a}$
are considered.
Let the integer $m_0\geqslant 0$ be the least possible such
that
the characteristic function $h({\mathfrak a},m)$ is
stable for $m\geqslant m_0$.
Then
either $m_0=b_0$, or $m_0=b_1-1$. More precisely, if $m_0=b_1-1$ then
$b_0=b_1$
and there is no pair $\langle h,u\rangle$ in a $0$-standard exact cone
decomposition
of $N_I$ with
$|u|=0$ and $\deg h=b_1-1$. On the other hand, by Lemma~7.2 \cite{5}
the
degrees of polynomials in the considered reduced Gr\"obner basis are bounded
from above by $b_0$. The assertion of Remark~1 follows from here.

Notice also that before Lemma~7.2 \cite{5} there is the following
formula: $b_0=\min\{d\geqslant b_1 :\forall_{z\geqslant
d}\,\overline{\varphi}(z)=\varphi(z)\}$. Here we have corrected the
misprint (which may mislead the reader now)
and replaced $z>d$ by $z\geqslant d$. Otherwise, this formula
contradicts to the definition of $b_i$ on p.765 of \cite{5}.


\par\medskip\noindent{\bf REMARK~2}\hspace{0.1em} {\it  Let ${\mathfrak
P}\subset A$ be an arbitrary homogeneous prime ideal
of height $\mbox{\rm
ht}({\mathfrak P})=s$, $0\leqslant s\leqslant n$, and of degree $d$.
We shall suppose additionally that the ideal $\overline{k}\otimes_k{\mathfrak
P}\subset\overline{k}\otimes_kA$ is radical
or which is equivalent the variety ${\cal Z}({\mathfrak P})$ is defined
over $k$.
Then by Lemma~9 and Corollary~1
the characteristic function $h({\mathfrak P},m)$
of this ideal is stable
for $m\geqslant\min\{(sd)^{O(n-s+1)^{n-s}},d^{2^{O(n)}}\}$ and hence
${\mathfrak P}$ has a
system of generators
with degrees at most $\min\{(sd)^{O(n-s+1)^{n-s}},d^{2^{O(n)}}\}$.
}\par\medskip



\medskip\noindent{\bf PROBLEM}\quad {\it Are there constants $c>0$,
$n_0>0$, $d_0>0$
satisfying the following property? For all integers $n>n_0$, $d>d_0$
for every field $k$
there is a homogeneous prime ideal
${\mathfrak p}\subset k[X_0,\ldots , X_n]$
with the degree $\deg{\mathfrak p}=d$ and the height $\mbox{\rm
ht}({\mathfrak p})=2$
such that for this ideal ${\mathfrak p}$ assertion (c) of Theorem~1 holds.}




\section{Lower bounds for the stabilization of the characteristic function
of a binomial ideal in arbitrary characteristic}\label{s1}

A power product in the ring
$k[X_1,\ldots , X_{n-1}]$
is a monomial $X_1^{i_1}\cdot\ldots\cdot X_{n-1}^{i_{n-1}}$ for some integers
$i_j\geqslant 0$.
By definition a binomial from $k[X_1,\ldots , X_{n-1}]$
is a difference of two distinct power products.
A polynomial ideal ${\mathfrak g}\subset k[X_1,\ldots ,X_{n-1}]$ is called
binomial
if and only if it has a system of generators $q_1,\ldots , q_v$ consisting
of binomials.
In the similar way binomials and binomial ideals are defined for the ring
$A^{(0)}=k[X_0,\ldots , X_{n-1}]$ and other polynomial rings.


\par\medskip\noindent{\bf LEMMA~1}\hspace{0.1em} {\it
There are constants $c_1>0$, $n_0>0$ and $d_0>0$ such
that for any field $k$
(of arbitrary characteristic and not necessarily infinite) for all integers
$n>n_0$, $d>d_0$ there is a homogeneous ideal
${\mathfrak b}\subset k[X_0,\ldots , X_{n-1}]=A^{(0)}$ satisfying the
following condition. The ideal ${\mathfrak b}$ is generated by a system of
homogeneous binomials
$b_1,\ldots , b_\mu$ of degrees $\deg b_i=m_i<d$
and $\mu=O(n)$.
Suppose that the
characteristic function $h({\mathfrak b},m)$ is stable
for $m\geqslant m_0$, i.e., coincides with the
Hilbert polynomial of $A^{(0)}/{\mathfrak b}$ for these $m$.
Then $m_0\geqslant
d^{2^{c_1n}}$.
}\par\medskip

\noindent{\bf PROOF}\quad Let us show, cf. \cite{11},
that the reduced Gr\"obner basis of any binomial ideal
with respect to any admissible ordering of monomials consists of binomials.
Indeed, one can construct this Gr\"obner basis by the Buchberger algorithm.
Each new polynomial in the Buchberger algorithm comes from
either (i) the reduction of a binomial by another binomial, or
(ii) by the $S$-polynomial of two binomials.
In both (i) and (ii) the result is again a binomial.
This proves the required assertion.

Besides that, if $e_1,\ldots , e_\alpha$ is the reduced Gr\"obner basis of
the binomial ideal
$(q_1,\ldots , q_v)\subset {\Bbb Q}[X_1,\ldots , X_{n-1}]$
with respect to an admissible ordering of monomials for any fixed
field $k$ of zero--characteristic (say, for  the field of
rational numbers $k={\Bbb Q}$) then by the same argument
$e_1\bmod p,\ldots , e_\alpha\bmod p$ is the reduced Gr\"obner basis
of the ideal $(q_1\bmod p,\ldots , q_v\bmod p)\subset k_p[X_1,\ldots ,
X_{n-1}]$
with respect to the same ordering of monomials over any
field $k_p$ of characteristic $p$ for every prime integer $p$.
Hence the number of elements $\alpha$ and the maximal degree
$\max_{1\leqslant i\leqslant\alpha}e_i$ of the reduced Gr\"obner basis
does not depend on the
choice of the field $k$, in particular of the characteristic of the field
$k$.

In \cite{11} the following assertion is proved
(actually with  ${\Bbb Q}[X_1,\ldots ,X_n]$ in place of
${\Bbb Q}[X_1,\ldots ,X_{n-1}]$ but for us it is
convenient to formulate it for ${\Bbb Q}[X_1,\ldots ,X_{n-1}]$).
There are constants $c_0>0$, $n_0>0$ and $d_0>0$ such that for all integers
$n>n_0$, $d>d_0$ there are an integer $\mu=O(n)$ and
binomials $g_1,\ldots , g_\mu\in {\Bbb Q}[X_1,\ldots ,X_{n-1}]$
(they are constructed explicitly) of degrees $\deg g_i<d$, $1\leqslant
i\leqslant\mu$, satisfying the following condition.
Put the ideal ${\mathfrak g}=(g_1,\ldots , g_\mu)\subset{\Bbb
Q}[X_1,\ldots ,X_{n-1}]$.
Let $e_1,\ldots , e_\alpha$ is the reduced Gr\"obner basis
of ${\mathfrak g}$ with respect
to some admissible ordering $<$ of monomials
(this ordering can be chosen arbitrary).
Then $\max_{1\leqslant i\leqslant\alpha}\deg e_i\geqslant d^{2^{c_0n}}$.
By the previous arguments the same assertion holds with an arbitrary field
$k$ in place of ${\Bbb Q}$. In what follows
we shall denote by $e_1,\ldots, e_\alpha$ the reduced Gr\"obner basis of the
ideal $(g_1,\ldots, g_\mu)\subset k[X_1,\ldots,X_{n-1}]$ with respect to the
ordering of monomials $<$.

For every $1\leqslant i\leqslant\mu$ put $b_i=X_0^{\deg
g_i}g_i(X_1/X_0,\ldots , X_{n-1}/X_0)\in A^{(0)}$
to be the homogenization of $g_i$. Hence $b_i$ is
a homogeneous binomial from
$k[X_0,\ldots, X_{n-1}]$. By definition put ${\mathfrak b}=(b_1,\ldots ,
b_\mu)\subset
k[X_0,\ldots , X_{n-1}]$
to be
the homogeneous ideal.
We shall assume without loss of generality that the ordering of monomials
$<$ is degree-compatible, i.e., for all monomial $v_1,v_2\in A$ the
condition
$\deg v_1<\deg v_2$ implies $v_1<v_2$. Further, consider the
admissible ordering of monomials from $A^{(0)}$ which extends the ordering
$<$ on
the monomials from
$k[X_1,\ldots ,X_{n-1}]$ and such that $X_0^i<X_j$ for all $1\leqslant
j\leqslant n$ and $i\geqslant 0$. Let $e'_1,\ldots , e'_\lambda$ be the
reduced Gr\"obner basis
of
the ideal ${\mathfrak b}$ with respect to the considered
ordering of the monomials. Then from the definition of the Gr\"obner basis
we get
immediately that $e'_1(1,X_1,\ldots ,X_{n-1}),\ldots ,
e'_\lambda(1,X_1,\ldots ,$ $X_{n-1})$ is a Gr\"obner basis with respect to
ordering
$<$ of the ideal $(g_1,\ldots , g_\mu)$.
Since the ordering $<$ on the monomials from $k[X_1,\ldots ,X_{n-1}]$ is
degree-compatible we
have
$$
\max_{1\leqslant i\leqslant\lambda}\deg e'_i(1,X_1,\ldots
,X_{n-1})\geqslant\max\deg_{1\leqslant i\leqslant \alpha} e_i\geqslant
d^{2^{c_0n}}.
$$
Therefore, $\max_{1\leqslant i\leqslant\lambda}\deg e'_i\geqslant
d^{2^{c_0n}}$.
Now by Remark~1 the assertion of the lemma holds. The lemma is proved.


\medskip In \cite{11} one can find also
a survey of the results on this subject. All the works here
are based on
the initial ideas from \cite{9} where zero--characteristic fields are
considered (or, more precisely, the field of rational numbers).
The proof of Lemma~2 \cite{9} about all possible binomials from a binomial
ideal is given for zero--characteristic.
D. Yu. Grigoriev (a private communication)
noticed that there is another simple proof of
this lemma in arbitrary characteristic.
It seems that there are no other obstacles to extend these results for
the case of nonzero characteristic.
Hence all the lower bounds for binomial
ideals are valid in arbitrary characteristic.
In our proof of Lemma~1 we even don't use Lemma~2 from \cite{9}
in nonzero characteristic.




\section{Proof of Theorem~1: conditions (a) and (b)}\label{s2}


The ring of polynomials $A=k[X_0,\ldots , X_n]$ is graded by the degree
with respect to all the variables $X_0,\ldots , X_n$.
For a graded $A$-mo\-du\-le $M$ and an integer $\nu$ denote by $M(\nu)$ the
graded $A$-mo\-du\-le
with the shifted graduation. Namely,
one can identify the homogeneous components $M(\nu)_m=M_{\nu+m}$ for every
$m$ and this identification induces the isomorphism of $A$-mo\-du\-les
$M(\nu)\simeq M$.
The last isomorphism is an isomorphism degree $\nu$ of graded
$A$-mo\-du\-les $M(\nu)$ and $M$.


Let $X_{n+1}$, $Z_1,\ldots , Z_\mu$ be new
variables. Let
$$
\Phi=X_0X_nX_{n+1}+X_n^3+X_{n+1}^3.
$$
Set the rings
of polynomials
$A^{(1)}=A[X_{n+1}]$. Put the ring
$B_\Phi=A^{(1)}/(\Phi)$.
Now $A$, $A^{(1)}$, $A^{(1)}[Z_1,\ldots , Z_\mu]$ are graded rings by
the total degree with respect
to all the variables
$X_0,\ldots,X_{n+1}$, $Z_1,\ldots, Z_\mu$,
i.e., the homogeneous
elements of degree $m$ these rings are
homogeneous polynomials of degree $m$ with respect to all these variables.
The polynomial $\Phi$ is homogeneous. Hence $B_\Phi$
and $B_\Phi[Z_1,\ldots , Z_\mu]$ are also a graded rings.

Denote
by $K^{(1)}$ and $K'$ the fields of fractions of
$A^{(1)}$ and $B_\Phi$ respectively. For an arbitrary
rational function $P/Q\in K^{(1)}$ such that $P,Q\in A^{(1)}$ and
$\Phi$ does not divide $Q$ we shall denote by $(P/Q)\bmod\Phi$ the
image of this rational function in $K'$.

\par\medskip\noindent{\bf LEMMA~2}\hspace{0.1em} {\it  Let $k$ be an
arbitrary field. Let ${\mathfrak a}\subset A$ be
a homogeneous ideal
such that $X_n\in{\mathfrak a}$ or ${\mathfrak
a}=A$.
Put
$$
B_{\mathfrak
a}=\left\{\left(z_0+z_1X_{n+1}+z_2\frac{X_{n+1}^2}{X_n}\right)\bmod\Phi\, :\,
z_0,z_1\in A\,\&\,z_2\in{\mathfrak a}\right\}\subset K'.
$$
Then $B_{\mathfrak a}$ is a graded ring
and a finitely generated $A$-mo\-du\-le.
Further, the $A$-mo\-du\-le $B_{\mathfrak a}\simeq A\oplus
A(-1)\oplus
{\mathfrak a}(-1)$.
 The ring $\overline{k}\otimes_kB_{\mathfrak a}$ is integral.
The field of fractions of $B_{\mathfrak a}$ is $K'$, the tensor product
$\overline{k}\otimes_kK'$ is a field, and
the extension of fields $\overline{k}\otimes_kK'\supset
\overline{k}(X_0,\ldots , X_n)$ is finite separable.
}\par\medskip

\noindent{\bf PROOF}\quad The element $X_{n+1}^2\bmod\Phi\in B_{\mathfrak
a}$ since $X_n\in{\mathfrak a}$.
For all $x_1,x_2\in{\mathfrak a}$ we have
\begin{eqnarray*}
&&x_1X_{n+1}^3/X_n\bmod\Phi=(-x_1X_n^2-x_1X_0X_{n+1})\bmod\Phi\in
B_{\mathfrak
a}, \\
&&x_1x_2X_{n+1}^4/X_n^2\bmod\Phi=
(-x_1x_2X_nX_{n+1}-x_1x_2X_0X_{n+1}^2/X_n)\bmod\Phi\in B_{\mathfrak a}
\end{eqnarray*}
and
$B_{\mathfrak a}$ is an $A$-mo\-du\-le. Hence $B_{\mathfrak a}$ is a
ring.
Other assertions are also straightforward.
The lemma is proved.

\medskip
Let ${\mathfrak b}\subset A^{(0)}$ be the ideal from Lemma~1.
We shall suppose without loss of generality that $n_0>1$ in Lemma~1 and
hence $n\geqslant 2$.
Set ${\mathfrak b}'={\mathfrak b}A+(X_n)$
to be the homogeneous ideal of $A$.
Set ${\mathfrak a}={\mathfrak b}'$ in Lemma~2 and $B'=B_{{\mathfrak
b}'}$.
We have $A/{\mathfrak b}'=A^{(0)}/{\mathfrak b}$.
Hence if $\dim (B')_m$ is stable for
$m\geqslant m_0$ (i.e., coincides with the
Hilbert polynomial of $A^{(0)}/{\mathfrak b}$
for these $m$) then $m_0\geqslant
d^{2^{c_1 n}}+1$ by Lemma~1.


\medskip We shall construct recursively the graded ring $B^{(0)}=B'$,
$B^{(1)},\ldots , B^{(\mu)}$
satisfying for all $i$ the following properties.
\begin{enumerate}\renewcommand{\labelenumi}{(\roman{enumi})}
\item The ring $\overline{k}\otimes_kB^{(i)}$ is integral
\item The extension of rings $B^{(i)}\supset A$ is integral.
\item Denote by $K^{(i)}$ the field of fractions of $B^{(i)}$. Then
the extension of fields
$\overline{k}\otimes_kK^{(i)}\supset\overline{k}(X_0,\ldots , X_n)$ is
finite separable.
\item For $i\geqslant 1$ there are nonzero
$\lambda_{i,1},\lambda_{i,2},\lambda_{i,3},\lambda_{i,4}\in k$ such that
the polynomial
\begin{eqnarray*}
&&\varphi_i=Z_i^{m_i+1}+\lambda_{i,1}X_0^{m_i}Z_i+\lambda_{i,2}X_0^{m_i}X_1+
\lambda_{i,3}X_0^{m_i}X_2+ \\
&&\lambda_{i,4}(b_iX_{n+1}^2/X_n)\bmod\Phi\in B^{(i-1)}[Z_i]
\end{eqnarray*}
is an irreducible element of the ring
$\overline{k}\otimes_kK^{(i)}[Z_i]$, and by definition the ring
$B^{(i)}=B^{(i-1)}[Z_i]/(\varphi_i)$.
\end{enumerate}


For $i=0$ these properties follow from the definition of the ring $B'$.
Let $1\leqslant i\leqslant\mu$ and
$B^{(i-1)}$ is constructed. Let us construct $B^{(i)}$.
Obviously (i)--(iii) follow from (iv).
So it is sufficient to construct the polynomial $\varphi_i$ from (iv).

Denote by $E^{(i-1)}$ the integral closure of the ring
$\overline{k}\otimes_kB^{(i-1)}$ in its
field of fractions $\overline{k}\otimes_kK^{(i-1)}$.
Notice that $E^{(i-1)}[Z_i]\supset\overline{k}\otimes_kB^{(i-1)}[Z_i]$.
Denote by $V^{(i-1)}$ the normal affine algebraic variety with the ring of
regular
functions $E^{(i-1)}[Z_i]$.

Let $\alpha\in k$ be a nonzero element.
Let us define the elements of the ring $\overline{k}\otimes_kB^{(i-1)}[Z_i]$
$$
\begin{array}{lll}
\psi_1=X_0^{m_i}X_2, &
\psi_2=X_0^{m_i}Z_i, &
\psi_3=X_0^{m_i}X_1,\\
\widetilde{\psi}_1=\alpha Z_i^{m_i+1}+X_0^{m_i}X_2, &
\widetilde{\psi}_2=X_0^{m_i}Z_i, \\
\widetilde{\psi}_3=X_0^{m_i}X_1+
\alpha(b_iX_{n+1}^2/X_n)\bmod\Phi.
\end{array}
$$
Then
$$
V^{(i-1)}\cap{\cal
Z}(\widetilde{\psi}_1,\widetilde{\psi}_2,\widetilde{\psi}_3)\subset
V^{(i-1)}\cap{\cal Z}(Z_i,X_0X_2).
$$
Hence by (ii) for $i-1$
$$
\dim (V^{(i-1)}\cap
{\cal
Z}(\widetilde{\psi}_1,\widetilde{\psi}_2,\widetilde{\psi}_3))\leqslant
\dim(V^{(i-1)})-2.
\eqno (1)
$$
Further, by (iii) for $i-1$ the family
$X_0,X_0^{m_i}X_1,X_0^{m_i}X_2,X_3,X_4,\ldots , X_n,X_0^{m_i}Z_i$
is a separable basis of transcendency of the field
$\overline{k}\otimes_kK^{(i-1)}(Z_i)$ over $\overline{k}$.
Therefore, the morphism
$$
V^{(i-1)}\rightarrow{\Bbb A}^3(\overline{k}),\quad
z\mapsto(\psi_1(z),\psi_2(z),\psi_3(z))
\eqno (2)
$$
is separable
dominant, or which equivalent the differential of this morphism at some
smooth point $\xi$ is an epimorphism

Let us show that there is a nonempty open in the Zariski topology subset
${\cal U}''\subset\overline{k}$
(obviously the number of elements $\#(\overline{k}\setminus{\cal
U}'')<+\infty$) such that
for all $\alpha\in {\cal U}''$ the rational
morphism
$$
V^{(i-1)}\rightarrow{\Bbb P}^2(\overline{k}), \quad
z\mapsto(\widetilde{\psi}_1(z):\widetilde{\psi}_2(z):\widetilde{\psi}_3(z))
\eqno (3)
$$
is separable dominant.
Indeed,
consider the morphism
$$
\widetilde{\psi}\, :\,
V^{(i-1)}\rightarrow{\Bbb A}^3(\overline{k}), \quad
z\mapsto(\widetilde{\psi}_1(z),\widetilde{\psi}_2(z),\widetilde{\psi}_3(z)).
$$
Let $V^{(i-1)}\subset{\Bbb A}^{n_1+2}(\overline{k})$, $n_1\geqslant n+1$,
and ${\Bbb A}^{n_1+2}(\overline{k})$ has coordinate
functions $Z_i,X_0,$ $\ldots ,X_{n_1}$. Let $h_1,\ldots ,
h_s\in\overline{k}[Z_i,X_0,\ldots
,X_{n_1}]$ be a system of local parameters of the algebraic variety
$V^{(i-1)}$ at the point $\xi$.
Then the differential of morphism (2) at the point $\xi$ is an
epimorphism if and only if
the differentials
$d_{\xi}h_1,\ldots , d_{\xi}h_s,d_{\xi}\psi_1,d_{\xi}\psi_2,d_{\xi}\psi_3$
are linearly independent over $\overline{k}$. Therefore, the differentials
$d_{\xi}h_1,\ldots , d_{\xi}h_s$,
$d_{\xi}\widetilde{\psi}_1,d_{\xi}\widetilde{\psi}_2,
d_{\xi}\widetilde{\psi}_3$ are linearly independent over $\overline{k}$ for
all  $\alpha$ from a
nonempty open in the Zariski topology subset of
${\cal U}''\subset\overline{k}$.
Hence morphism  $\widetilde{\psi}$ is dominant and
separable for all  $\alpha\in{\cal U}''$.
The natural morphism $\pi\, :\,{\Bbb
A}^3(\overline{k})\setminus\{0\}\rightarrow{\Bbb P}^2(\overline{k})$
is dominant separable. Morphism  (3) is equal to
$\pi\circ\widetilde{\psi}$.
Hence the last morphism is also separable dominant for all
$\alpha\in{\cal U}''$.
This proves the required assertion.

Let $V',V''\subset{\Bbb A}^n(\overline{k})$ be arbitrary affine algebraic
varieties. Recall that by definition
the intersection of  $V'$ and $V''$ is transversal
if and only if for every irreducible over $\overline{k}$
component $W$ of the intersection  $V'\cap V''$
there is a smooth point $z\in W$ such that $z$ is a smooth point of
$V'$ and $V''$
simultaneously, and the intersection of tangent spaces of the algebraic
varieties
$V'$ and $V''$ at the point $z$ is
transversal where
the tangent spaces are considered as subspaces of ${\Bbb
A}^n(\overline{k})$.

{\it In what follows unless it is not stated otherwise we shall suppose that
the field $k$ is infinite.}
Let us choose and fix $\alpha\in{\cal U}''\cap k$.
Recall that $V^{(i-1)}\subset{\Bbb A}^{n_1+2}(\overline{k})$.
The morphism  (3) is separable dominant, (1) holds and
$n\geqslant 2$.
Hence one can apply the first Bertini theorem, see  \cite{12},  \cite{1}, cf.
\cite{4}, to morphism (3).
By this theorem there is a polynomial $\varphi$ which is a
linear combination of
$\widetilde{\psi}_1,\widetilde{\psi}_2,\widetilde{\psi}_3$
with coefficients from $k$
in general position such that the intersection $\widetilde{V}^{(i-1)}$ and
${\cal Z}(\varphi)$
is transversal in ${\Bbb A}^{n_1+2}(\overline{k})$
and irreducible over
$\overline{k}$.

Since $E^{(i-1)}[Z_i]$ is integrally closed
for any nonzero element $\varphi'\in E^{(i-1)}[Z_i]$ the ideal
$(\varphi')\subset E^{(i-1)}[Z]$ is unmixed, i.e., all the associated prime
ideals of $(\varphi')$ have the same height $1$.

In our situation this implies that the ideal
$(\varphi)\subset E^{(i-1)}[Z_i]$ is prime. Hence the polynomial
$\varphi\in\overline{k}\otimes_kK^{(i-1)}[Z_i]$ is irreducible
since $E^{(i-1)}$ is integrally closed.
Multiplying $\varphi$ by a nonzero factor from $k$ we shall suppose
without loss of generality that
the leading coefficient with respect to $Z_i$ of the polynomial $\varphi$
is $1$. Now put $\varphi_i=\varphi$.
Hence condition (iv) is satisfied.
We get also $\lambda_{i,3}=\alpha^{-1}$, $\lambda_{i,4}=\alpha\lambda_{i,2}$
and hence
$\lambda_{i,2}=\lambda_{i,3}\lambda_{i,4}$.




Put $B=B^{(\mu)}$.
Denote by $K$ the field of fractions of $B$. Hence $\overline{k}\otimes_kK$
is the field of fractions of
$\overline{k}\otimes_kB$. By (iii) the extension
of fields $\overline{k}\otimes_kK\supset\overline{k}(X_0,\ldots , X_n)$
is finite separable.

By (iv) the isomorphism of graded $A$-mo\-du\-les $B\simeq
\oplus_{i\in I}B'(-\alpha_i)$ holds,
where the number of elements $\#I=(m_1+1)\cdot\ldots\cdot
(m_\mu+1)\leqslant d^\mu=d^{O(n)}$
and
$0\leqslant\alpha_i\leqslant
(m_1+\ldots + m_\mu)\leqslant(d-1)\mu=O(nd)$.

By (iv) the ring $B$ is generated over $k$ by its homogeneous component
$B_1$ and $\dim_k B_1\leqslant N+1$ where $N=n+1+\mu=O(n)$.
Put $X_{n+1+i}=Z_i$, $1\leqslant i\leqslant\mu$.
Let ${\Bbb P}^N(\overline{k})$ has the homogeneous coordinate functions
$X_0,\ldots , X_N$. Then by our construction
$\overline{k}\otimes_kB$ is a
homogeneous ring of a projective algebraic variety
$V'\subset{\Bbb P}^N(\overline{k})$ defined over $k$ and irreducible over
$\overline{k}$.
Set ${\mathfrak P}'$ to be the homogeneous ideal of the projective algebraic
variety $V'$.
Put $f_1=\Phi$ and
\begin{eqnarray*}
&&f_{i+1}=X_nX_{n+1+i}^{m_i+1}+\lambda_{i,1}X_0^{m_i}X_nX_{n+1+i}+
\lambda_{i,2}X_0^{m_i}X_1X_n+\\
&&\lambda_{i,3}X_0^{m_i}X_2X_n+
\lambda_{i,4}b_iX_{n+1}^2\in k[X_0,\ldots , X_N],\quad
1\leqslant i\leqslant\mu,
\end{eqnarray*}
cf. the formulas for $\varphi_i$ above.
Now  $B=k[X_0,\ldots, X_N]/{\mathfrak P}'$,
$$
f_1,\ldots , f_{\mu+1}\in k[X_0,\ldots , X_N],\quad
\deg f_i<d'=O(d),\,1\leqslant
i\leqslant\mu+1,\quad
N=O(n).
\eqno (4)
$$
Further, ${\cal Z}({\mathfrak P}')$ is a defined over $k$ and
irreducible over $\overline{k}$ component of ${\cal Z}(f_1,$ $\ldots ,
f_{\mu+1})$.  More precisely, we have
the decomposition into the union of two irreducible over $\overline{k}$
components: ${\cal Z}(f_1,\ldots ,
f_{\mu+1})={\cal Z}({\mathfrak P}')\cup{\cal Z}(X_n,X_{n+1})$.
Finally, by our construction
${\mathfrak P}'$ is a primary component of the
ideal $(f_1,\ldots ,f_{\mu+1})\subset k[X_0,\ldots , X_N]$,
the height $\mbox{\rm ht}({\mathfrak P}')=\mu+1$ and the
dimension $\dim{\cal Z}({\mathfrak P}')=n$.
Hence assertion (a) of the theorem holds for $N$,
$(f_1,\ldots , f_{\mu+1})$, ${\mathfrak P}'$ (in
place of $n$, $(f_1,\ldots , f_\nu)$, ${\mathfrak p}$).

Let $\alpha=\max_{i\in I}\alpha_i$.
Then $\dim_k B_m$ is stable for
$m\geqslant m_0$ if and only if $\dim_k B'(-\alpha)_m$ is stable for
$m\geqslant m_0$.
We have $B'\simeq A\oplus A(-1)\oplus {\mathfrak b}'(-1)$ and ${\mathfrak
b}'=A{\mathfrak b}+(X_n)$ where
${\mathfrak b}$ is the
ideal from Lemma~1.
Hence if $\dim_k B_m$ is stable for
$m\geqslant m_0$ then $\dim_k((A^{(0)}/{\mathfrak b})(-\alpha-1))_m$ is
stable for
$m\geqslant m_0$.
Therefore, $m_0\geqslant d^{2^{c_1n}}+\alpha+1$ by Lemma~1.


\par\medskip\noindent{\bf REMARK~3}\hspace{0.1em} {\it  Let ${\mathfrak
P}\subset A$ be a homogeneous prime ideal. Let
$A'=A[X_{n+1},\ldots ,$ $X_{n'}]$ for
$n'\geqslant n$.
Consider the homogeneous prime ideal ${\mathfrak P}'={\mathfrak
P}A'+(X_{n+1},\ldots
,$ $X_{n'})\subset A'$.
Then the Hilbert function of $A/{\mathfrak P}$ is stable for $m\geqslant
m_0$ if and only if
the Hilbert function of $A'/{\mathfrak P}'$ is stable for $m\geqslant m_0$.
Further, the ideal ${\mathfrak P}$ has a system of generators $b_1,\ldots ,
b_w$ with all
$\deg b_i\leqslant v$ if and only if the ideal ${\mathfrak P}'$ has a
system of generators
$b'_1,\ldots , b'_{w'}$ with all
$\deg b'_i\leqslant v$.

Now one can consider the prime ideal ${\mathfrak
P}''={\mathfrak P}A'$.
The Hilbert function of $A'/{\mathfrak P}''$ is stable for $m\geqslant
m_0$ if and only if
the Hilbert function of $A'/{\mathfrak P}'$ is stable for $m\geqslant
m_0+n'-n$ (this is proved using the induction on $n'-n$; here
$m_0$ or $m_0+n'-n$ may be negative).
}\par\medskip


The integers $n$, $d$ are arbitrary sufficiently big. Hence
from our construction and (4) using Remark~3 and
changing the notations ($N$, ${\mathfrak P}'$,
$d'$ by $n$, ${\mathfrak P}^{(0)}$,  $d$
respectively; we leave the details to the reader) we get the
following assertion.
\begin{itemize}
\item[(*)]
{\it
there are constants $c_2>0$, $n_0>0$, $d_0>0$ (they may be distinct from the
ones from Lemma~1)
such that for all integers $n>n_0$, $d>d_0$
for every infinite field $k$
there are homogeneous polynomials $f_1,\ldots , f_{\mu+1}\in k[X_0,\ldots ,
X_n]$, $\mu<n$, with all $\deg f_i<d$,
and a homogeneous prime ideal ${\mathfrak
P}^{(0)}\subset k[X_0,\ldots , X_n]$
satisfying conditions (a) and (b) of Theorem~1
(in place of the polynomials $f_1,\ldots , f_\nu$ and the prime ideal
${\mathfrak p}$)
with $c_2$ in place of $c$.
Besides that, according to our construction the height $\mbox{\rm
ht}({\mathfrak P}^{(0)})=\mu+1\leqslant n-1$.
The polynomials $f_1,\ldots , f_{\mu+1}$ depend on the choice of the
family of
elements $\lambda_{i,j}$, $1\leqslant i\leqslant\mu$, $1\leqslant j\leqslant
3$, see above.
Finally, ${\mathfrak P}^{(0)}$ is a primary component of the
ideal $(f_1,\ldots ,f_{\mu+1})\subset k[X_0,$ $\ldots , X_n]$, and the
projective
algebraic variety
${\cal Z}(f_1,\ldots , f_{\mu+1})$ has exactly two irreducible over
$\overline{k}$ components: ${\cal Z}({\mathfrak P}^{(0)})$ and some
linear subspace ${\cal L}^{(0)}\subset{\Bbb P}^n(\overline{k})$
defined over $k$.}
\end{itemize}


\section{Proof of Theorem~1: condition (c)}\label{s3}


Now our aim is to consider (a), (b) and (c) together.
\par\medskip\noindent{\bf REMARK~4}\hspace{0.1em} {\it  It would be
interesting
to ascertain that ${\mathfrak P}^{(0)}$ also satisfies (c)
with a constant $c_3>0$
in place of $c$, and after that take  $c=\min\{c_2,c_3\}$ (actually
$c_2\geqslant c_3$).
Since $3\mu=O(n)$ from here
one could deduce similarly to the proof of Proposition~1,
see below at the end of the section, that Theorem~1 is true also for any
finite field $k$,
i.e., without any restrictions on the field $k$.
}\par\medskip

But we shall obtain another result which is
sufficient to prove the theorem.

\medskip
Let us identify the set of all $(n-s)$-tuples $(L_{s+1},\ldots,L_n)$
of linear forms $L_j\in \overline{k}[X_0,\ldots , X_n]$, $s\leqslant
j\leqslant n$, with ${\Bbb A}^{(n+1)(n-s)}(\overline{k})$.

Let ${\mathfrak P}\subset A$
be an arbitrary homogeneous prime ideal of height $s=\mbox{\rm
ht}({\mathfrak P})$, $0\leqslant s\leqslant n-1$, and of degree
$\deg{\mathfrak P}=d$.
Let $V={\cal Z}({\mathfrak P})\subset{\Bbb P}^n(\overline{k})$ be the
projective algebraic variety of all common zeroes of
the polynomials from ${\mathfrak P}$ in ${\Bbb P}^n(\overline{k})$. Hence
$\dim V=n-s$.
We shall suppose additionally
that the ideal $\overline{k}\otimes_k{\mathfrak
P}\subset\overline{k}\otimes_kA$ is radical or
which is equivalent the variety $V$ is defined over $k$.
Denote by ${\mathfrak M}=(X_0,\ldots , X_n)$ the maximal
homogeneous prime
ideal of $A$.
The following lemma is known for the case when the algebraic variety
${\cal Z}({\mathfrak P})\subset{\Bbb P}^n(\overline{k})$
is irreducible over $\overline{k}$ (and actually this case is sufficient for
our aims).
Still we would like to prove it in the general case.

\par\medskip\noindent{\bf LEMMA~3}\hspace{0.1em} {\it  There is a nonempty
open in the Zariski topology subset ${\cal
U}\subset{\Bbb A}^{(n+1)(n-s)}(\overline{k})$
satisfying the following properties.
Let an $(n-s)$-tuple of linear forms
$(L_{s+1},\ldots, L_n)\in{\cal U}$ and all $L_j\in A$. Then
\begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})}
\item for
every integer $s\leqslant i\leqslant n$ the intersection $V$ and
${\cal Z}(L_{s+1},\ldots, L_i)$ is
transversal
or which is equivalent in the considered case the dimension
$\dim(V\cap{\cal
Z}(L_{s+1},\ldots, L_i))
=n-i$ and the degree
$\deg V=\deg(V\cap{\cal Z}(L_{s+1},\ldots, L_i))$;
\item for
every integer $s\leqslant i<n$ the number of irreducible over $\overline{k}$
components of $V_i=V\cap{\cal Z}(L_{s+1},\ldots ,L_i)$ is equal to the
number of irreducible over
$\overline{k}$ components of $V$;
\item for
every integer $s\leqslant i\leqslant n$
the algebraic variety $V_i$ is
defined over $k$;
\item for
every integer $s\leqslant i<n$ the projective algebraic variety $V_i$ is
irreducible over $k$ and
corresponds to a homogeneous prime ideal  ${\mathfrak P}_{L_{s+1},\ldots,
L_i}\subset k[X_0,\ldots , X_n]$ of height
$i$;
\item for $i=n$ the projective algebraic variety $V_n$ is a defined over $k$
finite number of
points in
${\Bbb P}^n(\overline{k})$ corresponding
to a homogeneous radical ideal ${\mathfrak P}_{L_{s+1},\ldots, L_n}\subset
k[X_0,\ldots , X_n]$;
\item hence the ideal $\overline{k}\otimes_k{\mathfrak P}_{L_{s+1},\ldots,
L_i}\subset\overline{k}\otimes_kA$
is radical for all $s+1\leqslant i\leqslant n$;
\item for every $s+1\leqslant i\leqslant n$
$$
{\mathfrak P}+(L_{s+1},\ldots, L_i)={\mathfrak P}_{L_{s+1},\ldots, L_i}\cap
{\mathfrak Q}_{L_{s+1},\ldots, L_i},
$$
where ${\mathfrak Q}_{L_{s+1},\ldots, L_i}$ is an ${\mathfrak M}$-primary
ideal or ${\mathfrak Q}_{L_{s+1},\ldots, L_i}=A$;
\item hence $X_j^N{\mathfrak P}_{L_{s+1},\ldots, L_i}$ $\subset{\mathfrak
P}+(L_{s+1},\ldots, L_i)$ for all $0\leqslant j\leqslant n$ and all
sufficiently big $N\geqslant 0$.
\end{enumerate}
}\par\medskip

\noindent{\bf PROOF}\quad There is a nonempty open in the Zariski topology
subset  ${\cal U}^{(0)}\subset{\Bbb A}^{(n-s)(n+1)}(\overline{k})$
such that for every $(L_{s+1},\ldots ,L_n)\in{\cal U}^{(0)}$ the
intersection $V$ and ${\cal Z}(L_{s+1},\ldots ,L_n)$ is transversal,
i.e., (i) holds (here we leave the details to the reader).
Let $\overline{k}\otimes_k{\mathfrak
P}=\bigcap_{j\in J}{\mathfrak p}_j$ be
the irredundant primary decomposition of the ideal
$\overline{k}\otimes_k{\mathfrak P}$, where all ${\mathfrak
p}_j\subset\overline{k}\otimes_kA$, $j\in J$, are
prime ideals. Then it is known (it is a corollary of the first Bertini
theorem),
see e.g., the Appendix of \cite{4},
that if one replaces ${\cal U}$, $k$, $A$, ${\mathfrak P}$ by ${\cal
U}_j$, $\overline{k}$,
$\overline{k}[X_0,\ldots , X_n]$, ${\mathfrak p}_j$
(for an arbitrary but fixed $j\in J$) respectively then assertion (i)--(v),
(vii), (viii) hold.
Put ${\cal U}={\cal U}^{(0)}\cap\bigcap_{j\in J}{\cal U}_j$.

Let $(L_{s+1},\ldots , L_n)\in{\cal U}$.
Now (i) and (ii) are satisfied.
The homogeneous ideal of the algebraic variety $V_i$ is $\bigcap_{j\in
J}{\mathfrak p}_{j,L_{s+1},\ldots , L_i}$.
We have $\overline{k}\otimes_k{\mathfrak P}+(L_{s+1},\ldots
,L_n)\overline{k}\otimes_kA\subset\bigcap_{j\in J}({\mathfrak
p}_j+(L_{s+1},\ldots ,L_n)\overline{k}\otimes_kA)$.
Hence by (vii) for ${\mathfrak p}_j$ the equality
$$
\overline{k}\otimes_k{\mathfrak P}+(L_{s+1},\ldots
,L_n)\overline{k}\otimes_kA=\bigcap_{j\in
J}{\mathfrak p}_j\cap{\mathfrak Q}'
\eqno (5)
$$
holds,
where ${\mathfrak Q}'$ is an  $\overline{k}\otimes_k{\mathfrak M}$-primary
ideal or  ${\mathfrak Q}'=\overline{k}\otimes_kA$.

Let us prove (iii). There is a nonzero linear form $L_0\in A$ such that
$V_i\setminus{\cal Z}(L_0)$
is a dense open in the Zariski topology subset in $V_i$.
By  (5)
 the ring of defined over $\overline{k}$ regular functions on
$V_i\setminus{\cal Z}(L_0)$ is
$\overline{k}[V\setminus{\cal Z}(L_0)]/(L_{s+1}/L_0,\ldots, L_i/L_0)$.
Hence $V_i\setminus{\cal Z}(L_0)$ is defined over
$k$.
Denote by $k_s$ the separable closure of $k$. Therefore, the set of points
$(V_i\setminus{\cal Z}(L_0))(k_s)$ is a
dense with respect to the Zariski topology and invariant with respect to the
action of the Galois group
$\mbox{\rm
Gal}(\overline{k}/k)$ subset of $V_i$.
Hence by the known criterion $V_i$ is defined over $k$. Assertion (iii) is
proved. Now also (v) holds.

Let us prove (iv). The Galois group
$\mbox{\rm
Gal}(\overline{k}/k)$ acts transitively
on the irreducible over $\overline{k}$ components ${\cal Z}({\mathfrak
p}_{j,L_{s+1},\ldots ,L_i})$ of the algebraic variety $V_i$.
Hence $V_i$ is irreducible over $k$ and (iv) is proved.
Now also (vi) is proved.

Let us prove (viii). By (5) we have
$$
X_j^N{\mathfrak P}_{L_{s+1},\ldots,
L_i}\subset(\overline{k}\otimes_k{\mathfrak
P}+(L_{s+1},\ldots, L_i)\overline{k}\otimes_kA)\cap A={\mathfrak
P}+(L_{s+1},\ldots, L_i)
$$
and (viii) is proved.
Obviously (viii) implies (vii). The lemma is proved.







\medskip Let $u=\{u_{i,j}\}$, $i\in\{0,s+1,s+2,\ldots , n+1\}$, $0\leqslant
j\leqslant n$,
be a family
of algebraically independent elements over the field $k$, i.e., the
transcendency degree of this family over
$k$ is $(n-s+2)(n+1)$. Denote by $k_u=k(u)$ the extension of the field $k$
by the elements of the family $u$.
Put $U_i=\sum_{0\leqslant j\leqslant n}u_{i,j}X_j\in k_u[X_0,\ldots , X_n]$,
$s+1\leqslant i\leqslant n$,
to be the family of generic linear forms over $k$.
Then the $(n-s)$-tuple $(U_{s+1},\ldots , U_n)$ is a generic point of
${\cal U}$, see Lemma~3,
and $(U_{n+1},\ldots , U_n)\in{\cal U}(\overline{k_u})$.
Let us extend the ground field $k$ till $k_u$.
We shall denote again by ${\mathfrak P}$ the ideal of
$k_u\otimes_k{\mathfrak P}\subset k_u\otimes_kA$ (this
will not lead to an ambiguity).
Now the ideals ${\mathfrak P}_{U_{s+1},\ldots , U_i}$ are defined
for all $s\leqslant i\leqslant n$.


\medskip Denote by $k[u]$ the polynomial ring with
coefficients from $k$ and variables from the family $u$.
Denote by $k[u,X_0,\ldots , X_n]$ the polynomial ring
coefficients from $k$ and variables
from the families $u$ and $X_0,\ldots , X_n$
(the similar notation will be used with other variables).
Let $s+1\leqslant i\leqslant n$ be an integer.
Denote for brevity ${\mathfrak P}_{i-1}={\mathfrak P}_{U_{s+1},\ldots ,
U_{i-1}}$,
${\mathfrak P}_i={\mathfrak P}_{U_{s+1},\ldots , U_i}$.
Put
$$
k''_u=\left\{\,P/Q\in k_u\,:\,P,Q\in k[u]\,\&\,\deg_{u_{i,n}}P\leqslant 0,\,
\deg_{u_{i,n}}Q=0\,\right\},
$$
i.e., $k''_u$ is the subfield of $k_u$ of all the elements which do not
depend on
$u_{i,n}$.
Notice that the ideal ${\mathfrak P}_{i-1}$ has a system of generators
$p_1,\ldots , p_\gamma\in k''_u[X_0,\ldots , X_n]$
(actually one can choose all $p_j\in k[u,X_0,\ldots , X_n]$ with
$\deg_{u_{i_1,j_1}}p_j=0$
for all $i_1\in\{0,i,\ldots , n+1\}$, $0\leqslant j_1\leqslant n$).
Let us define the multiplicatively closed sets
\begin{eqnarray*}
&&S_{{\mathfrak P}_{i-1}}=
k''_u[X_0,\ldots , X_n]\setminus(p_1,\ldots , p_\gamma)\subset
k''_u[X_0,\ldots , X_n], \\
&&S_{i,n}=k[u,X_0,\ldots , X_n]\setminus U_ik[u,X_0,\ldots , X_n]\subset
k[u,X_0,\ldots , X_n], \\
&&S=\{X_n^{i}\, :\, 0\leqslant i\in{\Bbb Z}\}\subset k[X_0,\ldots , X_n].
\end{eqnarray*}
The localization $S_{i,n}^{-1} k[u,X_0,\ldots , X_n]\supset k_u[X_0,\ldots ,
X_n]$.
If $z\in S_{i,n}^{-1} k[u,X_0,$ $\ldots ,X_n]$ then one can substitute
$u_{i,n}=(-\sum_{0\leqslant i\leqslant n-1}u_{i,j}X_j)/X_n$ in $z$.
Denote by $\pi(z)$ the result of this substitution.
Then obviously $\pi(z)\in k''_u(X_0,\ldots , X_n)$ and the mapping
$z\mapsto\pi(z)$ is
the ho\-mo\-mor\-ph\-ism $k''_u$-al\-ge\-b\-ras $\pi\, :\, S_{i,n}^{-1}
k[u,X_0,\ldots ,$ $X_n]\rightarrow k''_u(X_0,\ldots , X_n)$.
If $z\in k''_u(X_0,\ldots, X_n)$ then $\pi(z)=z$.
Obviously the kernel $\mbox{\rm Ker}(\pi)=U_iS_{i,n}^{-1}
k[X_0,\ldots , X_n]$.

\par\medskip\noindent{\bf LEMMA~4}\hspace{0.1em} {\it  Under previous
conditions suppose that  $X_n\not\in{\mathfrak P}$.
Then for every $z\in k_u[X_0,\ldots , X_n]$ the element $\pi(z)\in
S_{{\mathfrak P}_{i-1}}^{-1}
k''_u[X_0,\ldots , X_n]$.
Further, if $z\in{\mathfrak P}_i$ then $\pi(z)\in S_{{\mathfrak
P}_{i-1}}^{-1}(k''_u[X_0,\ldots , X_n]\cap{\mathfrak P}_{i-1})$.
Finally, if $z\in k[u,X_0,\ldots , X_n]\cap{\mathfrak P}_i$ then
$\pi(z)\in S^{-1} (k[u,X_0,\ldots , X_n]\cap{\mathfrak P}_{i-1})$.
}\par\medskip

\noindent{\bf PROOF}\quad Let $z\in k_u[X_0,\ldots , X_n]$. Then $z=P/Q$
where $0\ne Q\in k[u]$, $P\in k[u,X_0,\ldots , X_n]$.
We have $\pi(P)\in S^{-1} k''_u[X_0,\ldots , X_n]\subset S_{{\mathfrak
P}_{i-1}}^{-1}
k''_u[X_0,\ldots ,$ $X_n]$. Let $\deg_{u_{i,n}}Q=r$. Then $
X_n^r\pi(Q)\in k''_u[X_0,\ldots ,X_n]$.
Hence it is sufficient to prove that $
X_n^r\pi(Q)\in
S_{{\mathfrak P}_{i-1}}$.
The definition of the ho\-mo\-mor\-ph\-ism $\pi$
implies
$\pi(Q)=q((-\sum_{0\leqslant j\leqslant
n-1}X_ju_{i,j})/X_n)$
for a polynomial $0\ne q\in k''_u[Z]$.
Denote by $k_u({\cal Z}({\mathfrak P}_{i-1}))$ the field of defined over
$k_u$
rational functions on the variety ${\cal Z}({\mathfrak
P}_{i-1})\subset{\Bbb P}^n(\overline{k_u})$.
We have $X_n\not\in{\mathfrak P}$ and $U_{s+1},\ldots , U_{i-1}$ are generic
linear forms.
Hence $X_n$ does not vanish at the generic point of the
algebraic variety ${\cal Z}({\mathfrak P})\cap{\cal
Z}(U_{s+1},\ldots, U_{i-1})\subset{\Bbb P}^n(\overline{k_u})$.
Therefore,
$X_n\not\in{\mathfrak P}_{i-1}$ and obviously $i-1<n$.
Let
$$
k'''_u=\left\{\,P/Q\in k_u\,:\,P,Q\in k[u]\,\&\,\deg_{u_{i,j}}P\leqslant 0,\,
\deg_{u_{i,j}}Q=0,\,0\leqslant j\leqslant n \right\},
$$
i.e., $k'''_u$ is the subfield of $k_u$ of all the elements which do not
depend on all $u_{i,j}$, $0\leqslant j\leqslant n$.
The algebraic variety  ${\cal Z}({\mathfrak P}_{i-1})$ is
defined over $k'''_u$ by Lemma~3 with the ground field $k'''_u$ in place of
$k$.
Hence
$$
-\sum_{0\leqslant
j\leqslant n-1}u_{i,j}(X_j/X_n)\in k_u({\cal Z}({\mathfrak P}_{i-1}))
$$
is a
transcendental element over the field $k_u$.
Therefore,
$X_n^r\pi(Q)\not\in{\mathfrak P}_{i-1}$. The required
assertion is
proved.

Let $z\in{\mathfrak P}_i$. Then $z\in S^{-1}({\mathfrak P}_{i-1}+(U_i))$
by Lemma~3 (viii) with the ground field $k_u$ in place of $k$.
Hence one can represent $z=\sum_{1\leqslant j\leqslant \gamma}p_j
q_j+U_iq$ where
$p_j$ are introduced above, all $q_j,q\in S^{-1} k_u[X_0,\ldots ,
X_n]$.
Therefore,
$$
\pi(z)=\sum_{1\leqslant j\leqslant\gamma}p_j\pi(q_j)\in
S_{{\mathfrak P}_{i-1}}^{-1}(k''_u[X_0,\ldots , X_n]\cap{\mathfrak P}_{i-1}).
$$

Let $z\in k[u,X_0,\ldots , X_n]\cap{\mathfrak P}_{i-1}$.
Then by the proved
$$
\pi(z)\in S^{-1} k[u,X_0,\ldots , X_n]\cap
S_{{\mathfrak P}_{i-1}}^{-1}{\mathfrak P}_{i-1}=
S^{-1}(k[u,X_0,\ldots , X_n]\cap{\mathfrak P}_{i-1}).
$$
The last assertion and all the lemma are proved.




\par\medskip\noindent{\bf LEMMA~5}\hspace{0.1em} {\it  Under previous
notation suppose that for some  $s+1\leqslant
i\leqslant n-1$
the ideal  ${\mathfrak P}_{U_{s+1},\ldots , U_i}$ has a system of generators
of degrees with respect to $X_0,\ldots , X_n$ at most $D$ where $D\geqslant
2$.
Then the following assertions hold.
\begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})}
\item There is a system of generators
$q_1,\ldots , q_\beta\in k[u,X_0,\ldots ,
X_n]$
of the ideal ${\mathfrak P}_{U_{s+1},\ldots , U_i}$ such that
$\deg_{X_0,\ldots , X_n}q_j\leqslant D$ and
$\deg_{u_{v,w}}q_j=(Dd)^{O(n-i)}$ for all $v$, $w$
(here and below $\deg_{u_{w,v}}$ is the degree with respect to the variable
$u_{w,v}$) with an universal constant in $O(n-i)$.
\item There is a nonempty open in the Zariski topology subset
${\cal U}'_i\subset
{\cal U}$ satisfying the following properties.
There is an absolute constant $c'_4>0$
such that
for every $(L_{s+1},\ldots ,L_n)\in{\cal U}'_i$
the ideal
$$
X_j^N
{\mathfrak P}_{L_{s+1},\ldots , L_i}\subset{\mathfrak P}_{L_{s+1},\ldots ,
L_{i-1}}+(L_i)
\eqno (6)
$$
for an integer $N\leqslant(Dd)^{c'_4(n-i)}$.
Hence there is  an absolute constant $c_4>0$ such that
for every $(L_{s+1},\ldots ,L_n)\in{\cal U}'_i$
for all $m\geqslant(Dd)^{c_4(n-i)}$ the homogeneous components
$({\mathfrak P}_{L_{s+1},\ldots, L_i})_m$
and $({\mathfrak P}_{L_{s+1},\ldots,
L_{i-1}}+(L_i))_m$ (of the ideals ${\mathfrak P}_{L_{s+1},\ldots, L_i}$
and ${\mathfrak P}_{L_{s+1},\ldots,
L_{i-1}}+(L_i)$) coincide.
\end{enumerate}
}\par\medskip

\noindent{\bf PROOF}\quad Let us prove (i).
Consider the morphism $\pi_U\, :\,{\cal Z}({\mathfrak
P})\rightarrow{\Bbb P}^{n-s+1}(\overline{k_u})$, $(X_0:\ldots :
X_n)\mapsto(U_0:U_{s+1}:\ldots : U_{n+1})$
of projective algebraic varieties.
Then, cf. \cite{3}, the image
$\pi_U({\cal Z}({\mathfrak P}))$
is closed in ${\Bbb P}^{n-s+1}(\overline{k_u})$ and
defined over $k_u$, $\deg\pi_U({\cal Z}({\mathfrak P}))=d$.
Further,
$\pi_U({\cal Z}({\mathfrak P}))={\cal Z}(F_{\mathfrak P})$
where $F_{\mathfrak P}\in k[u,Z_0,Z_{s+1},\ldots ,$ $Z_n]$
is a homogeneous polynomial with respect to $Z_0,Z_{s+1},\ldots,$ $Z_n$
such that $F_{\mathfrak P}(U_0,U_{s+1},\ldots , U_{n+1})$ vanishes on
${\cal Z}({\mathfrak P})$ in ${\Bbb P}^n(\overline{k_u})$, the degree
$\deg F_{\mathfrak P}=d$ and the leading coefficient $0\ne\mbox{\rm
lc}_{Z_{n+1}}F_{\mathfrak P}\in k[u]$
(here we denote by $\mbox{\rm lc}_{Z_n}F_{\mathfrak P}$ the
leading coefficient of $F_{\mathfrak P}$ with respect to $Z_{n+1}$).
Since  $\pi_U({\cal Z}({\mathfrak P}))$ is defined over $k_u$ and
irreducible over $k_u$ the
polynomial $F_{\mathfrak P}$ does not have multiple factors
over $\overline{k_u}$ and is irreducible over $k_u$.
Since $0\ne\mbox{\rm
lc}_{Z_{n+1}}F_{\mathfrak P}\in k[u]$
the polynomial $F_{\mathfrak P}$ is separable with respect to
$Z_{n+1}$, i.e., $\partial F_{\mathfrak P}/\partial Z_{n+1}\ne 0$.
Besides that, cf. \cite{3},
the morphism ${\cal Z}({\mathfrak P})\rightarrow
\pi_U({\cal Z}({\mathfrak P}))$ induced by $\pi_U$ is a finite separable
and birational morphism of defined over $k_u$ projective algebraic varieties.

The polynomial $F_{\mathfrak P}$ is uniquely defined up to factor from
$k[u]$.
In what follows we shall assume without loss of generality that $\mbox{\rm
G\,C\,D\,}$ of all the coefficients from $k[u]$ at the
monomials in $X_0,\ldots , X_n$ of the polynomial $F_{\mathfrak P}$ is $1$.
So we fix
$F_{\mathfrak P}$ up to a nonzero factor from $k$.
We have $\deg_{u_{i,0},\ldots , u_{i,n}}F_{\mathfrak P}=d$ for every
$i\in\{0,s+1,s+2,\ldots , n+1\}$,
see  \cite{3} Lemma~9.


Let $L_0,L_{s+1},\ldots , L_{n+1}\in k[u,X_0,\ldots , X_n]$
be linear forms with respect to $X_0,\ldots , X_n$
in general position. Denote $L=(L_0,L_{s+1},\ldots,
L_{n+1})$.
Let $L_i=\sum_{0\leqslant j\leqslant n}l_{i,j}X_j$ where the
coefficients $l_{i,j}\in k_u$ for all $i,j$.
Let us substitute $u_{i,j}=l_{i,j}$ for all $i,j$ in $F_{\mathfrak P}$.
Denote the obtained polynomial by $F_{{\mathfrak P},L}\in k[u,X_0,\ldots ,
X_n]$.
Since $L_0,L_{s+1},\ldots , L_{n+1}$ are in general position
the polynomial $F_{{\mathfrak P},L}$ is separable with respect to $Z_{n+1}$,
$\deg F_{{\mathfrak P},L}=
\deg_{Z_{n+1}}F_{{\mathfrak P},L}=d$, $0\ne\mbox{\rm
lc}_{Z_{n+1}}F_{{\mathfrak P},L}\in k[u]$ and
the polynomial $F_{{\mathfrak P},L}(L_0,L_{s+1},\ldots, L_{n+1})$
vanishes on
${\cal Z}({\mathfrak P})$. Further, denote by $\pi_L\, :\,{\cal
Z}({\mathfrak
P})\rightarrow{\Bbb P}^{n-s+1}(\overline{k_u})$, $(X_0:\ldots :
X_n)\mapsto(L_0:L_{s+1}:\ldots : L_{n+1})$ the morphism
of projective algebraic varieties. Then the image
$\pi_L({\cal Z}({\mathfrak P}))$
is closed in ${\Bbb P}^{n-s+1}(\overline{k_u})$
and defined over $k_u$, $\deg\pi({\cal Z}({\mathfrak P}))=d$ and
$\pi_L({\cal Z}({\mathfrak P}))={\cal Z}(F_{{\mathfrak P},L})$.
Besides that, the morphism ${\cal Z}({\mathfrak P})\rightarrow
\pi_L({\cal Z}({\mathfrak P}))$ induced by $\pi_L$ is a finite separable
and birational morphism of defined over $k_u$ projective algebraic varieties.
In particular the polynomial $F_{{\mathfrak P},L}$ is irreducible over $k_u$.

Let $s\leqslant i\leqslant n-1$.
Put $L'=(L_0,U_{s+1},\ldots , U_i,L_{i+1},\ldots , L_{n+1})$ where all
$L_i\in k[X_0,\ldots , X_n]$ are linear forms
in general position. Recall the notation ${\mathfrak P}_i={\mathfrak
P}_{U_{s+1},\ldots , U_i}$.
Since $L_0$ is in general position we shall suppose in what follows
without loss of generality that
$L_0$ does not vanish on any irreducible over $\overline{k}$ component of
${\cal Z}({\mathfrak P}_i)$.
The polynomial $F'=F_{{\mathfrak P},L'}(Z_0,0,\ldots , 0,Z_{i+1},\ldots
, Z_{n+1})\in k_u[Z_0,Z_{i+1},\ldots , Z_{n+1}]$ (here we substitute $0$ for
$U_j$,
$s+1\leqslant j\leqslant i$)
is nonzero of degree $d$
since the leading coefficient of $F'$ with respect to
$Z_{n+1}$ is a nonzero polynomial from
$k[u]$. Obviously the polynomial $F'$ vanishes on ${\cal Z}({\mathfrak
P}_i)$.
Since linear forms $L_i$ are in general position the polynomial
$F_{{\mathfrak P}_i,(L_0,L_{i+1},\ldots , L_{n+1})}$ is defined and
satisfies the properties analogous to the ones of the polynomial
$F_{{\mathfrak P},L}$, see above.
Now $F_{{\mathfrak P}_i,(L_0,L_{i+1},\ldots , L_{n+1})}$
coincides with $F'$ up to a factor $f'\in k[u]$.
In particular the polynomial $F'$ is irreducible over $k_u$.

Let us replace $F'$ by $F'/f'$.
Denote by $k'_u$ the subfield of $k_u$ generated over $k$ by the
elements of the family $u'=\{u_{v,j}\}$, $s+1\leqslant v\leqslant i$,
$0\leqslant j\leqslant n$.
Now all the coefficients of $F'$ belong to $k'_u$

Put $t_j=L_j/L_0$, $i+1\leqslant j\leqslant n$, and $\Psi=F'(1,t_{i+1},\ldots
,t_n,Z)\in
k[u,t_{i+1},\ldots ,$ $t_n,Z]$. The polynomial  $\Psi\in k_u(t_{i+1},\ldots ,
t_n)[Z]$
is irreducible and separable with respect to $Z$, $\deg_Z\Psi=d$,
$0\ne\mbox{\rm lc}_Z\Psi\in k[u]$.
Hence the field of defined over $k_u$ rational functions
$k_u({\cal Z}({\mathfrak P}_i))\simeq k_u(t_{i+1},\ldots,
t_n)[Z]/(\Psi)$ according to the described construction.
Put
$$
\theta=Z\bmod\Psi\in
k_u(t_{i+1},\ldots, t_n)[Z]/(\Psi)\simeq k_u({\cal Z}({\mathfrak P}_i)).
\eqno (7)
$$
Obviously the coefficients of $\Psi$ belong to $k'_u$.

Put $L''=(L_0,U_{s+1},\ldots , U_i,L_{i+1},\ldots , L_n,U_{n+1})$
where the linear forms $L_{i+1},$ $\ldots, L_n$ be the as above.
Put $F''=F_{{\mathfrak P},L''}(Z_0,0,\ldots , 0,Z_{i+1},\ldots
, Z_{n+1})\in k_u[Z_0,$ $Z_{i+1},\ldots , L_n]$ (here we substitute $0$ for
$U_j$, $s+1\leqslant j\leqslant i$).
The rational function $F''(1,t_{i+1},\ldots , t_n,U_{n+1}/L_0)$ vanishes on
${\cal Z}({\mathfrak P}_i)\setminus{\cal Z}(L_0)$.
Hence there is a root $Z=\xi_U$ of the polynomial $F''(1,t_{i+1},\ldots,
t_n,Z)$ in the
field $k_u(t_{i+1},\ldots,t_n)[\theta]$ of the form
$$
\xi_U=\frac{1}{\xi^{(0)}}\sum_{0\leqslant v\leqslant
n}\left(\,\sum_{0\leqslant
j<\deg_Z\Psi}\xi_{v,j}\theta^j\,\right)u_{n+1,v},
$$
where all $\xi^{(0)},\xi_{v,j}\in k[u',t_{i+1},\ldots, t_n]$,
$0\leqslant j\leqslant n$,  $\xi^{(0)}\ne 0$, and
the greatest common divisor
$\mbox{\rm G\,C\,D\,}_{v,j}\{\xi^{(0)},\xi_{v,j}\}=1$
in the ring $k[u',t_{i+1},\ldots, t_n]$.

Thus, there is a generic point of the
algebraic variety
${\cal Z}({\mathfrak P}_{U_{s+1},\ldots, U_i})\subset{\Bbb
P}^n(\overline{k_u})$ of the form
$$
\frac{X_v}{L_0}=\xi_v=\frac{1}{\xi^{(0)}}\sum_{0\leqslant
j<\deg_Z\Psi}\xi_{v,j}\theta^j\in
k'_u(t_{i+1},\ldots , t_n)[\theta],\quad 0\leqslant
v\leqslant n,
\eqno (8)
$$
According to the algorithm for factoring polynomials, see  \cite{2}:
$\deg_Z\!\Psi$ $=d$, all the degrees
$$
\deg_{t_j}\Psi,\,
\deg_{u_{v,w}}\Psi,\,
\deg_{t_j}\xi^{(0)},\, \deg_{t_j}\xi_{v,j},\,
\deg_{u_{v,w}}\xi^{(0)},\, \deg_{u_{v,w}}\xi_{v,j}
$$
are bounded from above by
$(d+1)^{O(1)}$,
with an absolute constant in $O(1)$ for all $j$ $v,w$.






Denote by $I_\delta$ be the set of all $(i_0,\ldots , i_n)$ such that all
$i_w\geqslant 0$ are integers and
$i_0+\ldots + i_n=\delta$.
Let
$$
F=\sum_{(i_0,\ldots ,i_n)\in I_\delta}F_{i_0,\ldots ,
i_n}X_0^{i_0}\cdot\ldots\cdot
X_n^{i_n}
$$
be a homogeneous
polynomial of degree $\delta\leqslant D$ with arbitrary coefficients
$F_{i_0,\ldots,i_n}$ from the field $k_u$.
Then the relation
$$
F(\xi_0,\ldots ,\xi_n)=0
\eqno (9)
$$
 holds if and only if $F$
vanishes on ${\cal Z}({\mathfrak P}_{U_{s+1},\ldots , U_i})$.
Denote by $J_0$ the set of all
 $j=(j_{i+1},\ldots , j_n,j_0)$ such that all
$j_w\geqslant 0$ are integers and $0\leqslant j_0<\deg\Psi$.
Then by (8), (7) equality  (9) is equivalent to
$$
\sum_{j\in J_0}\left(\,\sum_{(i_0,\ldots ,i_n)\in I_\delta}a_{j,i_0,\ldots,
i_n}F_{i_0,\ldots , i_n}\,\right)t_{i+1}^{j_{i+1}}\cdot\ldots\cdot
t_n^{j_n}\theta^{j_0}=0,
\eqno (10)
$$
where all $a_{j,i_0,\ldots, i_n}\in k[u]$
and the greatest common divisor $\mbox{\rm
GCD}_{j,i_0,\ldots, i_n}\{a_{j,i_0,\ldots, i_n}\!\}$ $=1$
in $k[u]$. Further,
the ascertained bounds for degrees of $\xi_v$ and $\Psi$ imply that
for every nonzero $a_{j,i_0,\ldots, i_n}$ each index
$j_\alpha=(Dd)^{O(1)}$, $i+1\leqslant\alpha\leqslant n$, and the degrees
$\deg_{u_{v,w}}a_{j,i_0,\ldots, i_n}=(Dd)^{O(1)}$ for all $v,w$.
Hence there is a subset $J_1\subset J_0$ such that the
number of elements $\#J_1=(Dd)^{O(n-i)}$ and if $a_{j,i_0,\ldots, i_n}\ne 0$
then $j\in J_1$.
Therefore, considering the coefficients $F_{i_0,\ldots , i_n}$
as unknowns we get from
(10) a linear system $\sum_{(i_0,\ldots ,i_n)\in
I_\delta}a_{j,i_0,\ldots,
i_n}F_{i_0,\ldots , i_n}=0$, $j\in J_1$, with respect to
$F_{i_0,\ldots ,i_n}$
with coefficients $a_{j,i_0,\ldots, i_n}\in k_u$.
Solving this linear system we find a basis $y_1,\ldots , y_\gamma\in
k[u,X_0,\ldots , X_n]$
over the field $k_u$ of the
homogeneous component $({\mathfrak P}_{U_{s+1},\ldots , U_i})_\delta$ of
degree $\delta$ of the ideal
${\mathfrak P}_{U_{s+1},\ldots , U_i}$. According to the algorithm for
solving linear systems
we get $\deg_{u_{v,w}}y_j=(Dd)^{O(n-i)}$ for all $v$, $w$ and $1\leqslant
j\leqslant\gamma$.
Assertion (i) is proved.



Let us prove (ii).
Let $y=y_j$ for some $1\leqslant j\leqslant\gamma$, $\delta\leqslant D$.
We shall suppose without loss of generality performing if necessarily a
nondegenerate linear transformation of the coordinate functions
$X_0,\ldots ,X_n$ over $k$ that $X_i\not\in{\mathfrak P}$ for every
$0\leqslant i\leqslant n$ (after this
replacement the integer $N$ for old coordinate functions will
be $(n+1)N-n$).
Then, see Lemma~4, $\pi(y)\in S^{-1}({\mathfrak P}_{i-1}\cap
k[u,X_0,\ldots , X_n])$
and hence $y-\pi(y)\in U_iS^{-1} k[u,X_0,\ldots , X_n]$.
We have proved that $\deg_{u_{i,n}}y=(Dd)^{O(n-i)}$
with an absolute constant in $O(n-i)$.
Therefore, $X_n^N\pi(y)\in{\mathfrak P}_{U_{s+1},\ldots,
U_{i-1}}\cap k[u,X_0,\ldots, X_n]$ for an integer
$N=(Dd)^{O(n-i)}$. Hence $X_n^N(y-\pi(y))\in
U_ik[u,$ $X_0,\ldots, X_n]$. Thus $X_n^N{\mathfrak P}_{U_{s+1},\ldots ,
U_i}\subset
{\mathfrak P}_{U_{s+1},\ldots,
U_{i-1}}+(U_i)$. Similarly we have
$X_j^N{\mathfrak P}_{U_{s+1},\ldots , U_i}\subset
{\mathfrak P}_{U_{s+1},\ldots,
U_{i-1}}+(U_i)$ for every $0\leqslant j\leqslant n$.


Let $\delta_1=\delta+N$. We have constructed a basis $y_1,\ldots , y_\gamma$
of
the homogeneous component $({\mathfrak P}_{U_{s+1},\ldots , U_i})_\delta$
over the field $k_u$.
In the similar way we can construct a basis $y'_1,\ldots , y'_\sigma\in
k[u,X_0,\ldots , X_n]$ of
the homogeneous component $({\mathfrak P}_{U_{s+1},\ldots,
U_{i-1}})_{\delta_1}$.
Now for all $\rho,j$ one can represent
$$
X_\rho^Ny_j=\sum_{1\leqslant\alpha\leqslant\sigma}
b_{\rho,j,\alpha}y'_\alpha,\quad
b_{\rho,j,\alpha}\in k_u.
\eqno (11)
$$

To complete the proof we use ``the specialization of parameters'' $u_{v,j}$.
We leave to the reader to check that the described explicit
construction admits ``the specialization of parameters'',
i.e., if $(l_{v,j})$ belong to a
nonempty
open in the Zariski topology subset of ${\Bbb
A}^{(n-s+2)(n+1)}(\overline{k})$ then one can substitute
$u_{v,j}=l_{v,j}\in k$
in  (7), (8) and get
a generic point of the variety ${\cal Z}({\mathfrak P}_{L_{s+1},\ldots ,
L_i})$ with all
$L_v=\sum_{0\leqslant j\leqslant n}l_{v,j}X_j$.
Here Lemma~3 is also used.
Further, again for a nonempty
open in the Zariski topology subset one can substitute $u_{v,j}=l_{v,j}$  in
$y_1,\ldots , y_\gamma$ and
get the basis over the field $k$ of the
homogeneous component $({\mathfrak P}_{L_{s+1},\ldots , L_i})_\delta$ of
degree $\delta$ of the ideal
${\mathfrak P}_{L_{s+1},\ldots , L_i}$. This follows from the algorithm
for solving linear systems.
The similar assertion is fulfilled for the basis $y'_1,\ldots , y'_\sigma$
of the homogeneous component $({\mathfrak P}_{L_{s+1},\ldots,
L_{i-1}})_{\delta_1}$ of the ideal
${\mathfrak P}_{L_{s+1},\ldots , L_{i-1}}$.
For all $0\leqslant\delta\leqslant D$ formulas (11)
related  to generic point $(u_{v,i})$ also admit ``the specialization of
parameters''.
Finally, notice that we have the natural linear projection
\begin{eqnarray*}
&&\varepsilon\, :\,{\Bbb
A}^{(n-s+2)(n+1)}(\overline{k})\rightarrow{\Bbb
A}^{(n-s)(n+1)}(\overline{k}), \\
&&(L_0,L_{s+1},\ldots ,
L_{n+1})\mapsto(L_{s+1},\ldots , L_n),
\end{eqnarray*}
and for every open in the Zariski
topology subset of
${\cal W}$ of the affine space ${\Bbb
A}^{(n-s+2)(n+1)}(\overline{k})$ its
image $\varepsilon({\cal W})$ is
open in the Zariski topology in the affine space ${\Bbb
A}^{(n-s)(n+1)}(\overline{k})$.
So now using this projection $\varepsilon$ we get
the required subset ${\cal U}'_i$.
Thus, assertion (ii) and all the lemma are proved.

\par\medskip\noindent{\bf LEMMA~6}\hspace{0.1em} {\it  Under previous
notation let $s=\mbox{\rm
ht}({\mathfrak P})=n-1$.
Let ${\cal U}\subset{\Bbb A}^n(\overline{k})$
be the open in the Zariski topology subset of linear forms
corresponding to the prime
ideal ${\mathfrak P}$, see above.
Then for every $L_n\in{\cal U}$
the characteristic function $h({\mathfrak P}_{L_n},m)=\dim_k(A/{\mathfrak
P}_{L_n})_m$ is stable
for $m\geqslant(n-1)d-n+2$. Further,
for all $m\geqslant(n-1)d-n+2$  we have the equality
$$
({\mathfrak P}_{L_n})_m=({\mathfrak P}+(L_n))_m
\eqno (12)
$$
of homogeneous components of the degree $m$ of the ideals ${\mathfrak
P}_{L_n}$ and ${\mathfrak P}+(L_n)$.
}\par\medskip

\noindent{\bf PROOF}\quad There are homogeneous polynomials $F_1,\ldots
,F_m\in k[X_0,\ldots , X_n]$ of the same degree $d$ such that
${\cal Z}({\mathfrak P})={\cal Z}(F_1,\ldots ,F_m)$ in ${\Bbb
P}^n(\overline{k})$ and ${\mathfrak P}$ is a ${\mathfrak P}$-primary
component
of the ideal $(F_1,\ldots ,F_m)\subset k[X_0,\ldots , X_n]$,
cf.  \cite{2}, \cite{3}.
Then for a linear form $L_n\in{\cal U}$ the ideal ${\mathfrak
P}+(L_n)\supset(F_1,\ldots ,F_m,L_n)={\mathfrak
P}_{L_n}\cap{\mathfrak Q}$ where ${\mathfrak Q}$ is an
${\mathfrak M}$-primary ideal or ${\mathfrak Q}=k[X_0,\ldots , X_n]$.
Hence, see  \cite{8}, the homogeneous components $(F_1,\ldots
,F_m,L_n)_m=({\mathfrak P}_{L_n})_m$
of the ideals $(F_1,\ldots ,F_m,L_n)$ and ${\mathfrak P}_{L_n}$ coincide for
$m\geqslant(n-1)d-n+2$ and (12) holds.
Therefore, see  \cite{8},
the characteristic function $h({\mathfrak P}_{L_n},m)$ is stable
for $m\geqslant(n-1)d-n+2$,
cf. also Lemma~8 below. The lemma is proved.






\par\medskip\noindent{\bf LEMMA~7}\hspace{0.1em} {\it  Suppose that the
conditions of Lemma~5 hold for all $s+1\leqslant
i\leqslant n-1$
with the same $D$. Put ${\cal U}'=\bigcap_{s+1\leqslant
i\leqslant n-1}{\cal U}'_i$, see
Lemma~5 (ii) and
$$
D_1=\max\{(Dd)^{c_4(n-s-1)},sd-s\}.
$$
Then for every  $(L_{s+1},\ldots ,L_n)\in{\cal U}'$ for every
$s\leqslant i\leqslant n-1$ the characteristic function
$$
h({\mathfrak P}_{L_{s+1},\ldots , L_i},m)=\dim_k(A/{\mathfrak
P}_{L_{s+1},\ldots , L_i})_m
\eqno (13)
$$
 is stable for
$m\geqslant D_1$.
}\par\medskip

\noindent{\bf PROOF}\quad Let $(L_{s+1},\ldots ,L_n)\in{\cal U}'$.
Notice that
$k[X_0,\ldots, X_n]/(L_{s+1},\ldots, L_{n-1})$
is isomorphic to the ring of polynomials over $k$ in  $s+2$ variables
and ${\cal U}'\subset{\cal U}$ where ${\cal U}$
is an open set from Lemma~3.
Let us apply Lemma~6 to the ideal
$$
{\mathfrak P}_{L_{s+1},\ldots , L_{n-1}}/(L_{s+1},\ldots, L_{n-1})\subset
k[X_0,\ldots, X_n]/(L_{s+1},\ldots, L_{n-1})
$$
in place of ${\mathfrak P}\subset k[X_0,\ldots , X_n]$.
We get that for $i=n$ characteristic function (13)  is stable for
$m\geqslant sd-s+1$.
Now (12) and the exact sequence (14) with $i=n$,
see below, imply that
for $i=n-1$ characteristic function (13)  is stable for $m\geqslant
sd-s$.
This proves the required assertion for $i=n-1$.

We shall use the
decreasing induction on $i$. Suppose that
this characteristic function is stable for $m\geqslant D_1$ for some
$s+1\leqslant i\leqslant n-1$. Let us prove that it is stable for
$m\geqslant D_1$ also for $i-1$ (in place of $i$). We have exact sequence
$$
\begin{array}{l}
0\rightarrow (A/{\mathfrak P}_{L_{s+1},\ldots ,
L_{i-1}})_{m-1}\rightarrow
(A/{\mathfrak P}_{L_{s+1},\ldots , L_{i-1}})_m\rightarrow \\
(A/({\mathfrak P}_{L_{s+1},\ldots , L_{i-1}}+(L_i)))_m
\rightarrow 0
\end{array}
\eqno (14)
$$
of vector spaces and $(A/({\mathfrak P}_{L_{s+1},\ldots ,
L_{i-1}}+(L_i)))_m=(A/{\mathfrak P}_{L_{s+1},\ldots , L_i})_m$ for
$m\geqslant(Dd)^{c_4(n-s-1)}$ by Lemma~5. Hence
(13) is stable
for $m\geqslant D_1$ by the inductive assumption. The lemma is
proved.


\medskip\noindent{\bf CONCLUSION OF THE PROOF OF THEOREM~1}\quad
Put ${\mathfrak P}={\mathfrak P}_0$, see (*).
Hence now by (*) the height $\mbox{\rm ht}({\mathfrak
P})=s=\mu+1$ and by
the B\'ezout theorem
$\deg{\mathfrak P}=\deg{\mathfrak P}_0=d_1\leqslant d^s$
in place of $\deg{\mathfrak P}=d$.
We have
$s\leqslant n-1$ since $\mbox{\rm ht}({\mathfrak
P}^{(0)})\leqslant n-1$.
Let $D$ be the least integer such that the conditions of Lemma~5 hold
for all $s+1\leqslant i\leqslant n-1$ for
${\mathfrak P}={\mathfrak P}_0$
with $d_1$ in place of $d$.
Let ${\cal U}'_i$, $s+1\leqslant i\leqslant n-1$, be the sets from
Lemma~5 (ii) and
${\cal U}'=\bigcap_{s+1\leqslant i\leqslant n-1}{\cal U}'_i$.
Let us apply Lemma~7 with $d_1$ in place of $d$.
So now $D_1=\max\{(Dd_1)^{c_4(n-s-1)},sd_1-s\}$.
For $i=s$ by (*) we get $D_1\geqslant d^{2^{c_2n}}$.
Therefore, for all sufficiently big $n$ and $d$ the number
$D\geqslant d^{2^{c'n}}+1$ for an absolute constant $c'>0$.
Hence for every $(L_{s+1},\ldots ,L_n)\in{\cal U}'$ there is
$s+1\leqslant
i_0\leqslant n-1$ such that any system of generators
of the prime ideal ${\mathfrak P}_{L_{s+1},\ldots,L_{i_0}}$ contains
a polynomial of degree at least $d^{2^{c'n}}+1$
(the integer $i_0$ may depend on $(L_{s+1},\ldots ,L_n)$).
Let us choose and fix $(L_{s+1},\ldots ,L_n)\in{\cal U}'$.
Now we set ${\mathfrak p}={\mathfrak P}_{L_{s+1},\ldots,L_{i_0}}$ and
$c=c'$.
Hence ${\cal Z}({\mathfrak p})$ is a defined over $k$ and irreducible
over
$\overline{k}$ component of the algebraic variety
${\cal Z}(f_1,\ldots , f_{\mu+1},L_{s+1},\ldots,L_{i_0})$ by
Lemma~3.
Put $f_j=L_j$ for all $\mu+2\leqslant j\leqslant i_0$, and $\nu=i_0$. Now
all the polynomials
$f_1,\ldots , f_\nu$ are defined, and
assertion (a) of the theorem is satisfied.
Obviously assertion (c) holds. By Remark~1 also assertion (b)
is satisfied.
Theorem~1 is proved.

\medskip\noindent{\bf PROOF OF PROPOSITION~1}\quad
Let ${\mathfrak p}$ be the ideal constructed in the proof of Theorem~1.
Recall that in the proof we use the first Bertini theorem several times and
obtain
elements from $k$ in place of transcendental elements.
Now we would like to avoid applying the first Bertini theorem.
Namely, let us replace the family of elements $\lambda_{v,w}$, $1\leqslant
v\leqslant\mu$, $1\leqslant w\leqslant 3$
(recall that $\lambda_{v,4}=\lambda_{v,2}\lambda_{v,3}^{-1}$)
by the family $\lambda$ of transcendental over $k$ elements. We shall denote
them again by $\lambda_{v,w}$.
Denote by $k_\lambda$ the extension of the field $k$ by the all elements from
the family $\lambda$. So now the transcendency degree of the field
$k_\lambda$ over $k$ is
$3\mu$. Further, let us replace the family of coefficients of $l_{i,j}$,
$i\in\{0,s+1,\ldots , n+1\}$, $0\leqslant j\leqslant n$ of linear
forms $L_{s+1},\ldots , L_{n+1}$
by the family $u$ of transcendental over
$k_\lambda$ elements $u_{i,j}$ with transcendency degree $(n-s+2)(n+1)$
over $k_\lambda$.
Denote by $k_{\lambda,u}$ the extension of the field $k_\lambda$ by the all
elements
from
the family $u_{i,j}$.  So the transcendency degree of the field
$k_{\lambda,u}$ over $k$ is
$3\mu+(n-s+2)(n+1)=O(n^2)$.
Let $k$ be a finite field.
After this replacement by the construction from the
proof of Theorem~1 we get
a family of polynomials $f_1,\ldots , f_\nu\in k_{\lambda,u}[X_0,\ldots ,
X_n]$ (we use for them the same notation) and
a prime ideal ${\mathfrak p}'\subset
k_{\lambda,u}[X_0,\ldots , X_n]$
in place of ${\mathfrak p}\subset k[X_0,\ldots , X_n]$.

Denote by $k[\lambda,u,X_0,\ldots , X_n]$ the polynomial ring over $k$ with
the variables
from the families $\lambda$, $u$, $X_0,\ldots , X_n$. Put ${\mathfrak
p}''={\mathfrak p}'\cap k[\lambda,u,X_0,\ldots , X_n]$.
By the described construction all $f_1,\ldots, f_\nu\in
k[\lambda,u,X_0,\ldots, X_n]$ and all the degrees
$\deg_{\lambda,u,X_0,\ldots, X_n}f_i<d+1$.


The ideal ${\mathfrak p}''$ is not necessarily homogeneous with respect to
all the variables.
To simplify the notation denote all the variables from the families
$\lambda$, $u$, $X_0,\ldots , X_n$
by $Y_1,\ldots , Y_m$.
Denote by ${\Bbb A}^m(\overline{k})$ the affine space with the coordinate
functions $Y_1,\ldots , Y_m$.
The field $k$ is perfect since it is finite.
Hence ${\cal Z}({\mathfrak p}'')\subset{\Bbb A}^m(\overline{k})$ is a
defined over $k$ component of the algebraic variety ${\cal Z}(f_1,\ldots
, f_\nu)\subset{\Bbb A}^m(\overline{k})$.
The variety ${\cal Z}({\mathfrak p}'')\subset{\Bbb
A}^m(\overline{k})$
is irreducible over $\overline{k}$ since the variety ${\cal
Z}({\mathfrak p}')\subset{\Bbb P}^n(\overline{k_{\lambda,v}})$ is
irreducible over $\overline{k_{\lambda,v}}$ by the condition (a) of the
theorem.


Let $Y_0$ be a new variable. The homogenization of a polynomial $a\in
k[Y_1,$ $\ldots , Y_m]$ is the
polynomial $\overline{a}=Y_0^{\deg a}a(Y_1/Y_0,\ldots , Y_m/Y_0)\in
k[Y_0,\ldots , Y_m]$.
Let ${\mathfrak p}'''\subset k[Y_0,\ldots , Y_m]$ be the homogeneous prime
ideal
generated by all the elements $\overline{a}$, $a\in{\mathfrak p}''$.
Let ${\Bbb P}^m(\overline{k})$ be the projective space with the
homogeneous coordinate functions $Y_0,\ldots , Y_m$.
The projective algebraic varieties  ${\cal Z}({\mathfrak
p}''')\subset{\Bbb P}^m(\overline{k})$ and  ${\cal
Z}(\overline{f_1},\ldots ,\overline{f_\nu})\subset{\Bbb
P}^m(\overline{k})$
are closure with respect to Zariski topology of the affine
algebraic varieties
${\cal Z}({\mathfrak p}'')$ and ${\cal Z}(f_1,\ldots , f_\nu)$
respectively.
Therefore, ${\cal Z}({\mathfrak p}''')\subset{\Bbb
P}^m(\overline{k})$ is a defined over $k$
and irreducible over $\overline{k}$
component of the algebraic variety ${\cal Z}(\overline{f_1},\ldots
,\overline{f_\nu})\subset{\Bbb P}^m(\overline{k})$.
By (*) ${\mathfrak p}$ is primary component of the ideal $f_1,\ldots ,
f_\nu$ and the height $\mbox{\rm ht}({\mathfrak p})=\nu$.
Hence ${\mathfrak p}'''$ is primary component of the ideal
$(\overline{f_1},\ldots
,\overline{f_\nu})$ and the height $\mbox{\rm ht}({\mathfrak
p}''')=\nu$.
Thus, over the finite field $k$ assertion (a) of Theorem~1 holds for $m$,
$\overline{f_1},\ldots,\overline{f_\nu}$ and ${\mathfrak p}'''$ in place of
$n$, $f_1,\ldots , f_\nu$ and ${\mathfrak p}$ respectively.
Obviously all the degrees $\deg_{Y_0,\ldots , Y_m}\overline{f_i}<d+1$.
For the proof of the new versions of assertions (b) and (c),
see the statement of the proposition, we shall consider also
$\overline{f_1},\ldots,\overline{f_\nu}$ and ${\mathfrak p}'''$.

Assertion (c) of Theorem~1 holds for ${\mathfrak p}'$ over the field
$k_{\lambda,u}$ (in place of
${\mathfrak p}$ over the field $k$).
Any system of generators of the ideal ${\mathfrak p}''$
is a system of generators of the ideal ${\mathfrak p}'$.
Hence for any system of generators  $a_1,\ldots , a_m$ of the ideal
${\mathfrak p}''$
the maximal degree
$$
\max_{1\leqslant
i\leqslant
m}\deg_{Y_1,\ldots , Y_m}a_i\geqslant\max_{1\leqslant
i\leqslant
m}\deg_{X_0,\ldots , X_n}a_i\geqslant\deg d^{2^{cn}}
$$
Therefore, for any system of generators  $a'_1,\ldots , a'_{m'}$ of the
ideal ${\mathfrak p}'''$
$$
\max_{1\leqslant
i\leqslant
m'}\deg_{Y_0,\ldots , Y_m}a'_i\geqslant d^{2^{cn}}.
$$
This implies the assertion (c)
of the new version of the theorem,
see the statement of the proposition, since
$n>n_0$, $d>d_0$ are arbitrary, $m=O(n^2)$, all
$\deg\overline{f_i}<d+1$, see Remark~3.
Moreover, obviously here in
(c) one can replace  $d^{2^{c\sqrt{n}}}$ by
$d^{2^{c\sqrt{n}}}+1$ (and get a slightly more strong assertion).
Then the new version of assertion (b) follows from Remark~1.
The proposition is proved.

\par\medskip\noindent{\bf REMARK~5}\hspace{0.1em} {\it  Let $k$ be a finite
field and $t$ be a transcendental element over
$k$.
Let ${\mathfrak p}\subset k(t)[X_0,\ldots , X_n]$ be the ideal constructed
in the proof of Theorem~1
over an infinite field $k(t)$.
By the proof of Proposition~1 to prove Theorem~1 for the finite field $k$
it is sufficient to ascertain the following.
Applying the first Bertini theorem
one can choose all the coefficients $\lambda_{v,w}\in k[t]$ of the
polynomials
$f_{v+1}$,
and the coefficients $l_{i,j}\in k[t]$
of linear forms $L_i$ with degrees $\deg_t\lambda_{v,w}$, $\deg_tl_{i,j}$
bounded from above by $d^{n^{O(1)}}$.
}\par\medskip



\par\medskip\noindent{\bf REMARK~6}\hspace{0.1em} {\it
There is the similar problem also for an infinite field $k$.
Let the $\lambda_{v,w}\in k$ and
$l_{i,j}\in k$, see the proof of Theorem~1, correspond to the ideal
${\mathfrak p}$.
One needs to prove that the lengths of  $\lambda_{v,w}\in k$ and
$l_{i,j}\in k$ are bounded from above by $d^{n^{O(1)}}$.
}\par\medskip



One of the difficulties  related
to the assertions of Remark~5 and Remark~6 is to
estimate the size of the
normalization of an algebraic variety.
I could not find an explicit estimation of the normalization
in literature. But we see that one can give it.
We hope to return to this question in one of the next papers.
It seems that there are no other principal
difficulties, cf. the Appendix from \cite{4}.


\section{Upper bounds}\label{s4}


We are able to give also upper bounds for the stabilization of the
characteristic function of
a homogeneous polynomial prime ideal and for a system of generators of this
ideal. Let ${\mathfrak P}$  be the prime ideal from Section~3
but now we assume that the height $\mbox{\rm ht}({\mathfrak
P})=s$, $0\leqslant s\leqslant n$.
So the degree $\deg{\mathfrak P}=d$ and
the ideal $\overline{k}\otimes_k{\mathfrak
P}\subset\overline{k}\otimes_kA$
is radical.

\par\medskip\noindent{\bf LEMMA~8}\hspace{0.1em} {\it  Let ${\mathfrak
Q}\subset A$ be a homogeneous ideal such that the
ideal
$\overline{k}\otimes_k{\mathfrak
Q}\subset\overline{k}\otimes_kA$ is radical,
the dimension of the variety of zeroes $\dim{\cal Z}({\mathfrak q})=0$ in
${\Bbb P}^n(\overline{k})$ and the degree $\deg{\mathfrak Q}=d$.
Then the characteristic function $h({\mathfrak Q},m)$ is stable for
all $m\geqslant d-1$.
In particular, this is true for a homogeneous prime ideal ${\mathfrak
P}\subset A$ (in place of ${\mathfrak Q}$)
with $s=\mbox{\rm ht}({\mathfrak P})=n$ such that the algebraic
variety ${\cal Z}({\mathfrak P})\subset{\Bbb P}^n(\overline{k})$ is
defined over $k$.
}\par\medskip

\noindent{\bf PROOF}\quad Denote for brevity
$\overline{A}=\overline{k}\otimes_kA$.
We have $h({\mathfrak Q},m)=h(\overline{k}\otimes_k{\mathfrak Q},m)$.
Let $\overline{k}\otimes_k{\mathfrak Q}=\cap_{1\leqslant j\leqslant
d}{\mathfrak m}_j$ be the irredundant primary decomposition of
the ideal $\overline{k}\otimes_k{\mathfrak Q}$. Hence all ${\mathfrak
m}_j\subset\overline{A}$, $1\leqslant
j\leqslant d$, are homogeneous prime ideals of degree one and height $n$.
Hence it is sufficient to prove that the characteristic function
$h(\cap_{1\leqslant j\leqslant\delta}{\mathfrak m}_j,m)$ is stable for
$m\geqslant\delta-1$.
We use the induction on $\delta$.
The base $\delta=1$ is obvious. There is the exact sequence
of homogeneous components of degree $m$ induced by the exact sequence of the
ho\-mo\-mor\-ph\-isms
of graded $\overline{A}$-mo\-du\-les
$$
\begin{array}{l}
0\rightarrow (\overline{A}/(\cap_{1\leqslant
j\leqslant\delta}{\mathfrak m}_j))_m\rightarrow
(\overline{A}/(\cap_{1\leqslant j\leqslant\delta-1}{\mathfrak
m}_j))_m\times
(\overline{A}/({\mathfrak m}_\delta))_m\rightarrow \\
(\overline{A}/({\mathfrak m}_\delta+\cap_{1\leqslant
j\leqslant\delta-1}{\mathfrak m}_j))_m\rightarrow 0.
\end{array}
\eqno (15)
$$
There is a homogeneous polynomial $\varphi\in\cap_{1\leqslant
j\leqslant\delta-1}{\mathfrak m}_j\setminus{\mathfrak m}_\delta$ such that
$\deg\varphi=\delta-1$. Therefore,
$(\overline{A}/({\mathfrak
m}_\delta+\cap_{1\leqslant j\leqslant\delta-1}{\mathfrak m}_j))_m=\{0\}$
for $m\geqslant\delta-1$.
Hence by the inductive assumption and (15) the
characteristic function $h(\cap_{1\leqslant j\leqslant\delta}{\mathfrak
m}_j,m)$ is stable for
$m\geqslant\delta-1$. The lemma is proved.



\par\medskip\noindent{\bf LEMMA~9}\hspace{0.1em} {\it  Under the notation
from the beginning of the section suppose that
$1\leqslant s\leqslant n-1$.
Then the characteristic function $h({\mathfrak P},m)=\dim_k(A/{\mathfrak
P})_m$ is stable for all integers $m\geqslant (sd)^{(c_5(n-s))^{n-s-1}}$
for an absolute constant $c_5>0$.
Therefore, by Remark~1 the ideal ${\mathfrak P}$ has a system of generators
consisting of homogeneous polynomials of degrees at most
$1+(sd)^{(c_5(n-s))^{n-s-1}}$.
Finally, Lemma~8 now implies that for all $0\leqslant s\leqslant n$ the
characteristic function $h({\mathfrak P},m)$ is stable for
$m\geqslant (sd)^{(c_5(n-s+1))^{n-s}}$
and the ideal ${\mathfrak P}$ has a
system of generators
consisting of homogeneous polynomials of degrees at most
$1+(sd)^{(c_5(n-s+1))^{n-s}}$.
}\par\medskip

\noindent{\bf PROOF}\quad We shall suppose without loss of generality that
$1<s<n$, Hence $sd\geqslant 2$.
Let us show that
for all $(L_{s+1},\ldots ,L_n)$ from a nonempty open in the Zariski topology
subset of ${\Bbb A}^{(n-s)(n+1)}(\overline{k})$
for every $s\leqslant i\leqslant n-1$ the characteristic
function
(13)
is stable for $m\geqslant (sd)^{(c_5(n-s))^{n-i-1}}$,
and hence by Remark~1 the ideal ${\mathfrak P}_{L_{s+1},\ldots, L_i}$
has a system of homogeneous generators of degrees at most
$1+(sd)^{(c_5(n-s))^{n-i-1}}$ for an absolute constant $c_5>0$.
Indeed, this is true for  $i=n-1$ by Lemma~6.
We shall use the decreasing induction on $i$.
Suppose
that our assertion is proved for
some $s+1\leqslant i\leqslant n-1$.
Let us prove that it is true also for $i-1$ (in place of $i$).
We have the exact sequence
(14) of vector spaces. Hence by the
inductive assumption and Lemma~5
with $D=(sd)^{(c_5(n-s))^{n-i-1}}$
for all $(L_{s+1},\ldots,L_n)\in\bigcap_{i\leqslant
j\leqslant n-1}{\cal U}'_j$
(the set ${\cal U}'_i$ is defined at this
step of the induction in Lemma~5 (ii), and
similarly the sets ${\cal U}'_j$, $i+1\leqslant j\leqslant n-1$, are
defined at the previous steps of the induction)
the equality
$(A/({\mathfrak P}_{L_{s+1},\ldots ,
L_{i-1}}+(L_i)))_m=
(A/{\mathfrak P}_{L_{s+1},\ldots , L_i})_m$ holds
for
$$
m\geqslant(sd)^{(c_5(n-s))^{n-i}}\geqslant
((sd)^{(c_5(n-s))^{n-i-1}}d)^{c_4(n-i)}
$$
for an appropriate constant
$c_5$.
Therefore, characteristic function (13) with $i-1$ in place of $i$ is
stable
for
$m\geqslant(sd)^{(c_5(n-s))^{n-i}}$
by the inductive assumption.
The required assertion is proved.
The lemma is proved.



\par\medskip\noindent{\bf LEMMA~10}\hspace{0.1em} {\it  Under the notation
from the beginning of the section
there is an absolute constant $c_6>0$
such that
the prime ideal ${\mathfrak P}$ has a system generators
$q_1,\ldots , q_w\in k[X_0,\ldots , X_n]$
with $\deg q_i\leqslant d^{2^{c_6n}}$ for all $1\leqslant i\leqslant w$.
}\par\medskip

\noindent{\bf PROOF}\quad We can suppose without loss of generality that
$1<s<n$, see Lemma~8, and $d>1$.
There are homogeneous polynomials $F_1,\ldots
,F_m\in k[X_0,\ldots , X_n]$ of the same degree $d$ such that
${\cal Z}({\mathfrak P})={\cal Z}(F_1,\ldots ,F_m)$ in ${\Bbb
P}^n(\overline{k})$ and ${\mathfrak P}$ is a ${\mathfrak P}$-primary
component
of the ideal $(F_1,\ldots ,$ $F_m)\subset k[X_0,\ldots , X_n]$,
cf.  \cite{2}, \cite{3}. Let $\widetilde{F}_1,\ldots ,\widetilde{F}_s$ be
linear
combinations of $F_1,\ldots ,F_m$
with coefficients from $\overline{k}$ in general position.
Then by the first Bertini theorem, see  \cite{12},  \cite{1}, cf.
\cite{4}, the ideal of the ring
$\overline{k}[X_0,\ldots , X_n]$
$$
(\widetilde{F}_1,\ldots,\widetilde{F}_s)=(\overline{k}\otimes_k{\mathfrak
P})\cap\widetilde{{\mathfrak P}},
$$
where $\widetilde{{\mathfrak P}}$ is a homogeneous prime ideal of the ring
$\overline{k}[X_0,\ldots , X_n]$ with the height $\mbox{\rm
ht}(\widetilde{{\mathfrak P}})=s$,
the degree $\deg(\widetilde{{\mathfrak P}})=d^s-d>0$
and such that $\widetilde{{\mathfrak P}}$ is not a primary component of the
ideal
$\overline{k}\otimes_k{\mathfrak P}$.
Let $\overline{k}\otimes_k{\mathfrak P}=\bigcap_{j\in J}{\mathfrak p}_j$ be
the irredundant primary decomposition of the
radical ideal $\overline{k}\otimes_k{\mathfrak P}$.
There is a homogeneous polynomial $F\in\widetilde{{\mathfrak
P}}\setminus\bigcup_{j\in J}{\mathfrak p}_j$ such that $\deg F\leqslant
d^s-d$.
Now $\overline{k}\otimes_k{\mathfrak P}
=\{z\in\overline{k}[X_0,\ldots , X_n]\, :\,
zF\in(F_1,\ldots,F_m)\}$.
Consider the linear equation
$$
ZF=\sum_{1\leqslant i\leqslant s}Z_iF_i
\eqno (16)
$$
over the ring of polynomials $\overline{k}[X_0,\ldots , X_n]$.
By  \cite{6}, \cite{10} the submodule of $(\overline{k}\otimes_kA)^{s+1}$ of
solutions
$(Z,Z_1,\ldots ,Z_s)$ of equality (16) has a system of generators
$z_i,z_{i,1},\ldots , z_{i,s}$,
$i\in I$, consisting of polynomials from $\overline{k}\otimes_kA$ of degrees
$\deg z_i$, $\deg z_{i,j}$
bounded from above by $d^{2^{c_6n}}$ for an appropriate universal constant
$c_6>0$.
Now $z_i\in\overline{k}\otimes_kA$, $i\in I$,
is the system of generators of $\overline{k}\otimes_k{\mathfrak P}$ of the
required degrees.
Since the variety ${\cal Z}({\mathfrak P})$ is defined over $k$ there is
also a system of generators
$q_1,\ldots , q_w\in k[X_0,\ldots , X_n]$ of the ideal ${\mathfrak P}$
with $\deg q_i\leqslant d^{2^{c_6n}}$ for all $1\leqslant i\leqslant w$.
The lemma is proved.



\par\medskip\noindent{\bf COROLLARY~1}\hspace{0.1em} {\it  Under the
notation from the beginning of the section
there is an absolute constant $c_7>0$
such that the characteristic function $h({\mathfrak P},m)$ of
the prime ideal ${\mathfrak P}$ is stable for $m\geqslant d^{2^{c_7n}}$.
}\par\medskip

\noindent{\bf PROOF}\quad Indeed, by Lemma~8 and \cite{8}
we can suppose without loss of generality that $\mbox{\rm
ht}({\mathfrak P})=s\leqslant n-1$.
By Lemma~10 with the ground field $k_u$ (in place of $k$)
the conditions of Lemma~5 hold for all $s+1\leqslant i\leqslant n-1$ for
$D=d^{2^{c_6n}}+1$.
Let $(L_{s+1},\ldots , L_n)\in{\cal
U}'=\bigcap_{s+1\leqslant i\leqslant n-1}{\cal U}'_i$,
see Lemma~5 (ii).
Now the required
assertion follows from Lemma~7. The corollary is proved.



\newpage

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For this aim it is sufficient to
get an effective version of the first Bertini theorem (see  \cite{4}
where the case of zero--characteristic is considered)
for nonzero characteristic of the ground field.

\par\medskip\noindent{\bf REMARK~7}\hspace{0.1em} {\it  Under conditions of
Lemma~6 it is not difficult to prove (and it is
known) that
the characteristic function $h({\mathfrak P}_{L_n},m)$ is stable for
$m\geqslant d$.
Moreover, this is true for any radical ideal ${\mathfrak Q}\subset A$
(in place of ${\mathfrak P}$) such that $\overline{k}\otimes_k{\mathfrak
Q}\subset\overline{k}\otimes_kA$ remains radical,
the dimension of the variety of zeroes $\dim{\cal Z}({\mathfrak q})=0$ in
${\Bbb P}^n(\overline{k})$ and the degree $\deg{\mathfrak Q}=d$.
The proof is by the induction on $d$. We leave it to the reader.
In particular, this is true for a homogeneous prime ideal ${\mathfrak
P}\subset A$ (in place of ${\mathfrak Q}$)
with $s=\mbox{\rm ht}({\mathfrak P})=n$.
We shall use this remark only for the upper bounds with $s=n$ in Lemma~9,
Lemma~10 and Corollary~1.
}\par\medskip




\end{lems}
\noindent{\bf PROOF}\hspace{0.1em}
\end{rems}

%see, e.g.,  \cite{15},  \cite{16},  \cite{14},
