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%% Translation via Omnimark script a2l, September 19, 1996 (all in one day!)
\PII{S 1079-6762(96)00011-10}

%% Declarations:

\newtheorem*{theorem1}{Transformation Law}
\newtheorem*{theorem6}{Jacobi vanishing theorem}
\newtheorem*{theorem8}{Lemma 1}
\newtheorem*{theorem11}{Effective Nullstellensatz}

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\newcommand{\dist}{\operatorname {dist}}
\newcommand{\Fb}{\mathbf{ F}}
\newcommand{\missarg}{\,\cdot \,}
\newcommand{\N}{\mathbf{ N}}
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\newcommand{\Res}{\operatorname {Res}}
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\title{Residues and effective Nullstellensatz}
\author{Carlos~A. Berenstein}
\address{Institute for Systems Research, University of Maryland,
 College~Park, MD 20742}
\author{Alain Yger}
\address{Laboratoire de Math\'{e}matiques Pures,
 Universit\'{e} Bordeaux Sciences, 33405 Talence, France}
\thanks{This research has been partially supported by grants from NSA and
\keywords{Effective Nullstellensatz, residues, arithmetic B\'{e}zout
\subjclass{Primary 14Q20; Secondary 13F20, 14C17, 32C30}
\date{April 15, 1996}
\commby{Robert Lazarsfeld}
\begin{abstract}Let $\K $ be a commutative field; 
an algorithmic approach to residue
symbols defined on a Noetherian $\K $-algebra $\R $ has been developed.  It
is used to prove an effective Nullstellensatz for polynomials defined over
infinite factorial rings $\A $ equipped with a size.  This result extends
(and slightly improves) the previous work of the authors in the case
$\A =\Z $.

  Let $p_{1},\dotsc ,p_{M}$ be polynomials in $n$ variables with 
coefficients in
an integral domain $\A $, and respective degrees $D_{1}\geq D_{2}\geq \dotsb 
\geq D_{M}$,
with no common zeros in an integral closure of the quotient field $\K $ of
$\A $.  It follows from the versions of the Hilbert Nullstellensatz
in~\cite{B}, \cite{CGH}, \cite{Ko} that one can find an element
$r_{0}\in \A \setminus \{0\}$ and polynomials $q_{j}\in \A [x]$ such that
 \begin{equation*}r_{0}=\sum _{j=1}^{M} q_{j} p_{j}
 with a priori estimates on the degrees
 \begin{equation*}\max _{j}\deg q_{j}\leq (3/2)^{\iota }D_{1}\dotsm D_{\mu 
 where $\mu =\min \{n,M\}$ and $\iota =\max \{j\,:\,1\leq j<\mu -1,\ 
D_{j}=2\}$.  When
$\A =\Z $, using analytic methods, and mainly integral representation
formulas and multidimensional residues in $\C ^{n}$, one can show
\cite[Section~5]{BGVY} that system~(1) can be solved with the
 \begin{equation*}\begin{cases}\displaystyle \max _{j}\deg q_{j}\leq n(2n+
1)(3/2)^{\iota }\prod _{j=1}^{\mu }D_{j},\\
\displaystyle \max h(q_{j})\leq \kappa (n)D_{1}^{4}
\bigg (\prod _{j=1}^{\mu }D_{j}\bigg )^{8} (h+\log M+D_{1}\log 
 for the Mahler size $h(q_{j})$ in terms of the maximal Mahler size $h$ of the
original polynomials $p_{j}$. Recall that the Mahler size of 
$p\in \Z [x_{1},\dots ,x_{n}]$ is given by
\begin{equation*}h(p)=\int _{[0,1]^{n}}\log |p(e^{2\pi i\theta _{1}},\dots 
,e^{2\pi i\theta _{n}})|\,d\theta \,.
For $\A =\Z $, one can also recover the estimate for $\log \abs {r_{0}}$,
implicit in~(2), from the Arithmetic B\'{e}zout Theorem
(see~\cite{Ph2},\cite[Theorem~5.4.4]{BGS}), which shows that the 
Faltings height $H$ of the intersection of the arithmetic cycles $X_{j}$ in
$\Pb ^{n}(\Z )$ corresponding to the polynomials ${}^{h}p_{j}$ (homogeneous
versions of the original polynomials) has the bound
 \begin{equation*}H\leq c_{n} \Hc \biggl (\prod _{j=1}^{\nu }D_{j}\biggr )
\biggl (1/\Hc +\sum _{j=1}^{\nu }1/D_{j}\biggr ),\end{equation*}
 for some constant $c_{n}$, where $\Hc :=\max _{j} H(X_{j})$ and
$\nu :=\min \{n+1,M\}$.

  The origin of the results we announce here was the desire to obtain
estimates similar to the Effective Nullstellensatz (2) for factorial
rings $\A $ equipped with a size (in the sense of~\cite{Ph1}) and
simultaneously eliminate Analysis from the proof of such results. Apart 
from the case where $\A =\Z $ and the size is the Mahler size defined 
above, the natural example of such a ring is 
$\A =\Fb _{p}[\tau _{1},\dots ,\tau _{q}]$, where $p$ is a prime number and 
the size 
is the total degree in the parameters $\tau $.  The main
idea is to keep the structure of the proof in~\cite{BY1}, while eliminating
all the {\em analytic artifacts}.  In this process we develop the
computational aspects for a Residue Calculus, where the residues are
defined purely algebraically, following \cite{H} and, especially,
\cite{L}.  We are sure that this Residue Calculus will have many
applications beyond the scope of this paper, in particular, the algorithms
obtained permit the computation of residue symbols in the polynomial case,
without appealing to elimination theory or using Gr\"{o}bner bases.  In
particular, one should expect interesting geometric consequences of a
generalized form of the Jacobi vanishing theorem obtained here, as one
knows  in the classical case
\cite{G},\cite{Ku}.  The reader will find complete
proofs of these results in~\cite{BY2}.

\section*{1. Residue calculus}Let $\R $ be a Noetherian $\K $-algebra, 
where $\K $ is a commutative
field. A sequence $h_{1},\dots ,h_{n}$ generating an ideal $I$ in $\R $ is 
to be quasiregular \cite{M} if whenever one has a relation of the form
\begin{equation*}\sum _{k\in \N ^{n}:|k|=p}r_{k}h_{1}^{k_{1}}\dotsm 
h_{n}^{k_{n}}\in I^{p+1}\,, 
r_{k}\in \R \,,
for some $p\in \N $, then all the $r_{k}\in I$.
$x_{1},\dotsc ,x_{n}, h_{1},\dotsc ,h_{n}$ in $\R $ such that $(h_{1},\dotsc 
,h_{n})$ is 
a quasiregular sequence
generating the ideal $(h)=I$ and $\Pb :=\R /I$ is a finite-%
dimensional $\K $-vector space, we follow Lipman \cite[Chapter~3]{L} to
define the residue symbols
 as traces of certain $\K $-linear operators from $\Pb $ into itself. 
Namely, let $\E ={\mathrm{Hom}}_{\K }(\Pb ,\Pb )$ and $\sigma :\Pb 
\rightarrow \R $
be  a $\K $-linear 
section of the quotient map; then one can associate to any $Q\in \R $
the  element 
$Q^{\sharp }:=\sum _{k\in \N ^{n}}q_{k}^{\sharp }h^{k}\in \E [[h]]$, where 
the operators 
$q_{k}^{\sharp }\in \E $ are defined by the relations
\begin{equation*}Q\sigma (t)=\sum _{k\in \N ^{n}}\sigma (q_{k}^{\sharp 
}(t))h^{k}\,,t\in \Pb \,,\end{equation*}
in the $I$-adic completion of $\R $. 
Consider now the formal expansion in $\E [[h]]$,
\begin{equation*}Q^{\sharp }\circ \det \left ({\partial x_{i}^{\sharp }/ 
\partial h_{j}}\right ) 
=\sum _{k\in \N ^{n}} \delta _{k} h^{k}\,,
where the determinant is understood to take into account the non%
commutativity of the operator product. Although the operators 
$\delta _{k}$ depend on the choice of the section $\sigma $, their traces 
do not.  Now we can define the residue symbols as
 \begin{equation*}\Res \begin{bmatrix}Qdx_{1}\wedge \dotsb \wedge dx_{n}\\
h_{1}^{k_{1}+1},\dotsc ,h_{n}^{k_{n}+1}\end{bmatrix}
=\Res \begin{bmatrix}Qdx\\h^{k+\underline{1}}\end{bmatrix}
:={\mathrm{T}r}(\delta _{k}),
\quad Q\in \R ,\ k\in \N ^{n}\,.\end{equation*}
This definition extends the one given by Grothendieck \cite{H}. In the 
polynomial case, computations of residues are usually performed by taking as 
a section $\sigma $ the remainder with respect to a Gr\"{o}bner basis of 
$I$. One can also show
that in the analytic case (that is, $\R =\C [x_{1},\dotsc ,x_{n}]$ or $\R 
=\Oc $,
the ring of germs of holomorphic functions about $0\in \C ^{n}$) this
definition of residue coincides with the usual one, defined, for instance,
using Dolbeault cohomology \cite{GH}.  One can also see this using the
following Transformation Law \cite[Corollary~2.8]{L}.

  Let $f=(f_{1},\dotsc ,f_{n})$ and $g=(g_{1},\dotsc ,g_{n})$ be two 
sequences in $\R $, such that $g=Af$, where $A$ is an $n\times n$ matrix
with coefficients in $\R $, and such that the quotients $\R /(f)$ and
$\R /(g)$ are finite-dimensional $\K $-vector spaces. Let 
$\Delta $ be the determinant of the matrix $A$; then,
 for any
$x_{1},\dotsc ,x_{n},Q\in \R $, 
 \begin{equation*}\Res \begin{bmatrix}Qdx_{1}\wedge \dotsb \wedge 
dx_{n}\\f_{1},\dotsc ,f_{n}\end{bmatrix}
=\Res \begin{bmatrix}Q\Delta dx_{1}\wedge \dotsb \wedge dx_{n}\\
g_{1},\dotsc ,g_{n}\end{bmatrix}

  Extending the base ring $\R $ with transcendental parameters, we can
derive an extension of the Transformation Law.

  Let $f=(f_{1},\dotsc ,f_{n})$ and $g=(g_{1},\dotsc ,g_{n})$ be two 
sequences in $\R $, such that
 \begin{equation*}g_{j}=\sum _{l=1}^{n} a_{jl}f_{l},\quad j=1,\dotsc 
 where the coefficients $a_{jl}$ are in $\R $ and we let $\Delta $ be the
determinant of the matrix $A=[a_{jl}]$.  Then, for any
$x_{1},\dotsc ,x_{n},Q\in \R $ and any $k\in \N ^{n}$, 
 \begin{equation*}\Res \begin{bmatrix}Qdx\\f^{k+\underline{1}}\end{bmatrix}
=\sum _{\substack{\abs {q_{;j}}=k_{j}\\1\leq j\leq n}}\prod _{i=1}^{n}
\begin{pmatrix}\mu _{i}\\q_{i;}\end{pmatrix}
\Res \begin{bmatrix}Q\Delta \prod _{1\leq i,j\leq n}(a_{ij})^{q_{i,j}}dx\\
g_{1}^{\mu _{1}+1},\dotsc ,g_{n}^{\mu _{n}+1}\end{bmatrix}
 where we have introduced the following notation for the matrix of indices
$q_{i,j}\in \N $:
 \begin{equation*}q_{;j}=(q_{1,j},\dotsc ,q_{n,j}),\ q_{i;}=(q_{i,1},\dotsc 
\quad \mu _{i}=\abs {q_{i;}},\end{equation*}
 \begin{equation*}\begin{pmatrix}\mu _{i}\\q_{i;}\end{pmatrix}
=\mu _{i}!/q_{i,1}!\dotsm q_{i,n}!.\end{equation*}

  This proposition was stated in the analytic case in~\cite{Ky}.  The proof
given there is not complete.  One can also prove a very useful variant of
the Transformation Law, which we have not been able to find in the

  Let $f_{0},f_{1},\dotsc ,f_{n}$ be a regular sequence in $\R $.  Let
$g_{1},\dotsc ,g_{n}$ in $\R $ be such that the sequence $(f_{0},g_{1},\dotsc 
,g_{n})$ is 
quasiregular.  Assume that there are nonnegative integers $s_{1},\dotsc 
and an $n\times n$ matrix $A$ of elements in $\R $ such that 
 \begin{equation*}f_{0}^{s_{j}}g_{j}=\sum _{l=1}^{n} a_{jl}f_{l},\quad 
j=1,\dotsc ,n.\end{equation*}
 Let $x_{0},\dotsc ,x_{n},Q$ be elements of $\R $; then, for any $k_{0}\in 
\N $ one
 \begin{equation*}\Res \begin{bmatrix}Qdx_{0}\wedge \dotsb \wedge dx_{n}\\
f_{0}^{k_{0}+1},f_{1},\dotsc ,f_{n}\end{bmatrix}
=\Res \begin{bmatrix}Q\Delta dx_{0}\wedge \dotsb \wedge dx_{n}\\
f_{0}^{k_{0}+1+\abs {s}},g_{1},\dotsc ,g_{n}\end{bmatrix}
 where $\abs {s}=s_{1}+\dotsb +s_{n}$ and $\Delta $ is the determinant of the
matrix $A$.

  In the case where $\R =\K [x_{1},\dotsc ,x_{n}]$, which is our main interest
here, one can extend the residue symbol to act on rational functions, as
follows.  Given a quasiregular sequence $P_{1},\dotsc ,P_{n}$ in $\R $, let
$Q_{1}/Q_{2}$ be a rational function such that the ideal $(P_{1},\dotsc 
is $\K [x_{1},\dotsc ,x_{n}]$.  Namely, we define
 \begin{equation*}\Res \begin{bmatrix}(Q_{1}/Q_{2})dx_{1}\wedge \dotsb 
\wedge dx_{n}\\
P_{1},\dotsc ,P_{n}\end{bmatrix}
:=\Res \begin{bmatrix}{Q_{1} V}dx_{1}\wedge \dotsb \wedge dx_{n}\\
P_{1},\dotsc ,P_{n}\end{bmatrix}
 where $V$ is any polynomial such that for some $U_{1},\dotsc ,U_{n}$ in $\R $
one has $1=U_{1} P_{1}+\dotsb +U_{n} P_{n}+VQ_{2}$.  (This definition does 
not depend
on the choice of $V$.)

  In the previous transformation laws, one can replace $Q$ by a rational
function, provided the two residue symbols involved can be defined.

  From now on, we deal only with the case $\R =\K [x_{1},\dotsc ,x_{n}]$, 
$\K $ is a field of arbitrary characteristic.  Recall that a polynomial map
$P=(P_{1},\dotsc ,P_{n})$ from $\K ^{n}$ into itself is dominant if and only 
$\K (x)$ is a finite-dimensional $\K (P)$-vector space.  It is proper if and
only if $\K [x]$ is a finite type $\K [P]$-module.

  When $P$ is dominant, the residue symbol
 \begin{equation*}\Res \begin{bmatrix}Q(x)dx\\P_{1}-u_{1},\dotsc 
 (where $u_{1},\dotsc ,u_{n}$ are $n$ independent transcendental parameters) 
an element of $\K (u)$.  When $P$ is proper, this symbol is a polynomial in
$u$.  In order to obtain more information about this polynomial, we shall
assume that $\K $ is infinite, algebraically closed, and equipped with a 
absolute value $\abs {\missarg }$.  We  consider the norm defined
 on $\K ^{n}$  by
 $ \abs {x}=\max _{1\leq i\leq n}\abs {x_{i}}.$
 Recall that one can check properness by means of inequalities, namely, the
map $P$ is proper if and only if there exist three constants
$K,\gamma ,\delta >0$ such that
 \begin{equation*}\abs {x}\geq K \implies \abs {P(x)}\geq \gamma \abs 
{x}^{\delta }.
 Any $\delta >0$ for which (3) holds is called a {\em \L {}ojasiewicz
exponent\/} for $P$.

  Let $P\in \K [x_{1},\dotsc ,x_{n}]$.  We will denote as ${}^{h}P$ the 
polynomial in $n+1$ variables corresponding to the polynomial $P$ in
$\K [X_{0},\dotsc ,X_{n}]$, namely
 \begin{equation*}{}^{h}P(X_{0},\dotsc ,X_{n}):=X_{0}^{\deg P} 
P(X_{1}/X_{0},\dotsc ,X_{n}/X_{0}).\end{equation*}
 As a consequence of the Lipman-Teissier theorem \cite{LT} one has the
following result.

  Let $P=(P_{1},\dotsc ,P_{n})$ be a proper polynomial map in $\K ^{n}$ such 
condition $(3)$ holds for constants $K,\gamma ,\delta $, with
$\delta \in \N ^{*}$.  Assume that $D=\deg P_{j}$ for every $j$.  Then, for 
$1\leq j\leq n$, one can find a homogeneous polynomial $\Rc _{j}$ in two 
with coefficients in $\K $, distinguished in $X_{j}$, such that
 \begin{equation*}\big (\Rc _{j}(X_{0},X_{j})X_{0}^{D-\delta }\big )^{n+1}
\in I({}^{h}P_{1},\dotsc ,{}^{h}P_{n}),\end{equation*}
 the homogeneous ideal generated by the ${}^{h}P_{j}$.

  This proposition, combined with the Transformation Law and residue
calculus in one variable, leads to the following vanishing theorem for
residue symbols. 

  Let $P_{1},\dotsc ,P_{n}$ be polynomials in $\K [x_{1},\dotsc ,x_{n}]$ 
with $\deg P_{j}=D$, for $1\leq j\leq n$, which satisfy $(3)$ with an 
integral exponent
$\delta $.  Assume that
 \begin{equation*}(1-\epsilon _{n})D<\delta ,\quad \text{for }\epsilon 
 Then, for any $(k_{1},\dotsc ,k_{n})\in \N ^{n}$, one has
 \begin{equation*}\deg Q\leq n(n+1)(\abs {k}+n)(\delta -(1-\epsilon 
\implies \Res \begin{bmatrix}Qdx\\P^{k+\underline{1}}\end{bmatrix}
 Moreover, under the stronger hypothesis that
 \begin{equation*}(1-\epsilon _{n}/(n+1))D<\delta ,\end{equation*}
 one has
 \begin{equation*}\left .\begin{split}
&\deg Q\leq n(D-1)\\&k\not =0\end{split}\right \}
\implies \Res \begin{bmatrix}Qdx\\P^{k+\underline{1}}\end{bmatrix}

  The case where $D=\delta $ corresponds to the situation where the $P_{j}$
have no common zeros at $\infty $.  In this case, (4) is a particular
case (here all the polynomials have the same degree) of the Jacobi vanishing
theorem.  For the convenience of the reader, let us recall this theorem
here \cite{KK}.

  Let $P_{1},\dotsc ,P_{n}$ be elements of $\ \K [x_{1},\dotsc ,x_{n}]$ such 
that the
homogeneous parts of highest degree define a power of the maximal ideal
$I(x_{1},\dotsc ,x_{n})$.  Then
 \begin{equation*}\deg Q\leq \sum _{j=1}^{n}\deg P_{j}-n-1
\implies \Res \begin{bmatrix}Qdx\\P_{1},\dotsc ,P_{n}\end{bmatrix}

  Using analytic methods, we were able to obtain in~\cite{BY1}
a result
similar to~Proposition~4, without any restriction on the quotient
$\delta /D$.  That result can be stated as follows.  If $P$ is a proper
polynomial map from $\C ^{n}$ to $\C ^{n}$ such that the degrees $D_{j}$ of 
polynomials $P_{j}$ are in decreasing order and $\delta $ is a \L {}ojasiewicz
exponent, then, for any polynomial $Q\in \C [x_{1},\dotsc ,x_{n}]$ and 
$k$, one has
 \begin{equation*}(\abs {k}+2n-1)\delta >\deg Q+D_{1}+\dotsb +D_{n-1}+n
\implies \Res \begin{bmatrix}Qdx\\P^{k+\underline{1}}\end{bmatrix}
 This statement was crucial in the proof of the effective Nullstellensatz
over $\C $ given in~\cite{BY1}.  On the other hand, one can see that this
vanishing theorem is not the best one could expect.  For example,
(5) implies that
 \begin{equation*}\deg Q<(2n-1)\delta -(D_{1}+\dotsb +D_{n-1})-n
\implies \Res \begin{bmatrix}Q dx\\P_{1},\dotsc ,P_{n}\end{bmatrix}
 But a more careful analysis of the Bochner-Martinelli representation of
the residue currents yields the statement
 \begin{equation*}\deg Q\leq n\delta -n-1
\implies \Res \begin{bmatrix}Qdx\\P_{1},\dotsc ,P_{n}\end{bmatrix}
 The point here is that this result depends on the \L {}ojasiewicz exponent
$\delta $, but not on the degrees of the $P_{j}$.  We do not know how to prove
such result when $\K $ has positive characteristic, though it is possibly 
true.  Nevertheless, Proposition~4, which holds for any characteristic, is
enough to prove the effective Nullstellensatz theorem below.  We use it now
to obtain a global version of the Kronecker interpolation formula.

  Let $P_{1},\dotsc ,P_{n}$ be $n$ polynomials in $\K [x_{1},\dotsc ,x_{n}]$ 
of degree
$D$, with the property that there exist strictly positive constants
$K,\gamma $ such that $(3)$ holds for some integer $\delta >0$
 \begin{equation*}1-1/n(n+1)^{2}<\delta /D\leq 1.\end{equation*}
 Suppose that  $g_{jl}$,  $1\leq j,l\leq n$, are elements in
$\K [x_{1},\dotsc ,x_{n},y_{1},\dotsc ,y_{n}]$, with degree less or equal 
than $D-1$,
and such that
 \begin{equation*}P_{j}(x)-P_{j}(y)=\sum _{l=1}^{n} (x_{l}-y_{l})g_{jl}(x,y),
\quad 1\leq j\leq n,\ x,y\in \K ^{n}.\end{equation*}
 Then, if $\Delta (x,y):=\det [g_{jl}(x,y)]_{\substack{{1\leq j\leq 
n}\\{1\leq l\leq n}}}$ (such
$\Delta $ is called a {\em B\'{e}zoutian\/} for the map $P$), the following
polynomial identity holds:
 \begin{equation*}1=\Res \begin{bmatrix}\Delta (x,y)dx\\P_{1}(x),\dotsc 
\quad y\in \K ^{n}.\end{equation*}

  The transformation laws are also used to compute the residue symbols
 \begin{equation*}\Res \begin{bmatrix}Qdx\\P^{k+\underline{1}}\end{bmatrix}
 from the relations of integral dependence of the coordinates over $\K (P)$
when $P$ is a dominant map.  For instance, one can use the following lemma.

  Let $P_{1},\dotsc ,P_{n}$ be a quasiregular sequence in $\K [x_{1},\dotsc 
then for any $q\in \N $ and for any $\alpha \in \K ^{n}$ the sequence
$(t^{q+1},P_{1}(x)-\alpha _{1} t,\dotsc ,P_{n}(x)-\alpha _{n} t)$ is a 
sequence in $\K [x,t]$.  Moreover, we have the formula
 \begin{equation*}\Res \begin{bmatrix}Q(x)dt\wedge dx\\
t^{q+1},P_{1}(x)-\alpha _{1} t,\dotsc ,P_{n}(x)-\alpha _{n} t\end{bmatrix}
=\sum _{\abs {k}=q}\Res \begin{bmatrix}Qdx\\P^{k+\underline{1}}\end{bmatrix}
\alpha ^{k}.

  The left-hand side of (6) can be computed using relations of the
=\sum _{l=1}^{n} A_{j}^{l}(x_{j},P,u)(P_{l}-u_{l}), \quad j=1,\dotsc 
 so that it can be rewritten as
 \begin{equation*}t^{s_{j}}(R_{j}(x_{j},\alpha )-tS_{j}(x_{j},\alpha ,t))
=\sum _{l=1}^{n} A_{j}^{l} (x_{j},P,\alpha t)(P_{l}-\alpha _{l} t),
\quad R_{j}\not \equiv 0.\end{equation*}
 Then, we
use~Proposition~2 and residue computations in one variable, to compute the
left-hand side of (6).  Note that such expression is, in principle, an
element of $\K (\alpha _{1},\dotsc ,\alpha _{n})$, say $r_{1}(\alpha 
)/r_{2}(\alpha )$,
while we know from (6) that it is a polynomial in $\alpha $.  If the
$P_{j}$ have coefficients in some integral domain $\A $ (with quotient field
$\K $) equipped with a size,  then $r_{1}(\alpha )$ and $r_{2}(\alpha )$ have
coefficients in $\A $ and a common denominator for all residue symbols 
on the right-hand side of (6)
 divides $r_{2}(\alpha )$ in $\A [\alpha ]$, that is, it divides all the
coefficients of $r_{2}(\alpha )$.  This remark is crucial with respect to size
estimates for numerators or denominators of residue symbols.  It is clear
also that a good control on the size of the polynomials $A_{jl}$, for
$j=1,\dotsc ,n$, $0\leq l\leq N_{j}$, and $A_{j}^{l}$, for $1\leq j,l\leq n$ 
(which can be
chosen with coefficients in $\A $ in this case)  will provide good
estimates for the sizes of $r_{1}(\alpha )$ and $r_{2}(\alpha )$.

\section*{2. \L {}ojasiewicz inequalities}Another important tool in our 
proof of the Nullstellensatz is the {\em \L {}ojasiewicz inequality\/} 
\cite{JKS}.  It will be used in the following
form.  Given $n$ integers $D_{1}\geq D_{2}\geq \dotsb \geq D_{n}\geq 1$ we 
define, as
 \begin{equation*}B:=B(D_{1},\dotsc ,D_{n})=(3/2)^{\iota }D_{1}\dotsm 
 where $\iota =\#\{j0$ such that for any $x\in \K ^{n}$
 \begin{equation*}\abs {P(x)}
\geq \gamma \biggl (\dfrac{\min \{1,\dist (x,V(P))\}}{1+\abs {x}}\biggr 
 where $V(P)$ denotes the set of common zeros of the polynomials $P_{j}$ and
the distance corresponds to the absolute value $\abs {\missarg }$.

  Such an inequality, which is directly related to Koll\'{a}r's work
\cite{Ko} on the Nullstellensatz, has the following consequence, based on
the arguments given in \cite{BY1} and \cite[Propositions 5.7 and
5.8]{BGVY} when $\K =\C $.  Small modifications are required by the fact 
that we
are now working with fields of arbitrary characteristic.

  Let $P_{1},\dotsc ,P_{n}$ be a quasiregular sequence in $\K [x_{1},\dotsc 
then one can find $n$ linear combinations (with coefficients in $\K $) of
the $P_{j}$, namely, $\tilde {P}_{j}$, for $1\leq j\leq n$, and $n$ linearly
independent $\K $-linear forms $L_{1},\dotsc ,L_{n}$, as well as a positive
constant $K$ such that for any $N\in \N ^{*}$ and any $x\in \K ^{n}$ with
$\abs {x}>K$ one has
 \begin{equation*}\max _{1\leq j\leq n}\abs {L_{j}(x)}^{NB}\abs {\tilde 
\geq \gamma _{N} \abs {x}^{(N-1)B}\end{equation*}
 for some constant $\gamma _{N}>0$.

  In fact, the linear combinations $\tilde {P}_{j}$, as well as the linear
forms $L_{j}$, can be chosen with generic coefficients.  Moreover, if we 
combine this result with Proposition~5 and the Arithmetic B\'{e}zout Theorem
\cite[Theorem~4]{Ph1}, we get the following technical but important

  Let $P_{1},\dotsc ,P_{n}$ be a quasiregular sequence in $\K [x_{1},\dotsc 
with $D_{1}\geq D_{2}\geq \dotsb \geq D_{n}$, with $D_{j}:=\deg P_{j}$.  
Then one can find a
polynomial $\Phi $ in $n(n+1)+n^{2}$ variables $u_{jl},v_{jk}$, for
$1\leq j,k\leq n$, and $0\leq l\leq n$, with coefficients in $\K $, with
$\deg \Phi \leq 2^{n+1}(n+1)^{4} D_{1}^{n}$, such that, for any
$(U,V)\in \K ^{n\times (n+1)}\oplus \Mc _{n}(\K )$, with $\Phi (U,V)\neq 0$, 
 \begin{equation*}\Pi ^{j}_{U,V}(x):=U^{j}(x)\scalar {V^{j},P(x)}
:=\biggl (u_{j0}+\sum _{l=1}^{n} u_{jl}x_{l}\biggr )
\biggl (\sum _{l=1}^{n} v_{jl}P_{l}\biggr )\end{equation*}
 have degree exactly $D_{1}+1$, define a quasiregular sequence in
$\K [x_{1},\dotsc ,x_{n}]$, and moreover, if $N\in \N ^{*}$ is such that
 then the following polynomial identity holds in $\K [x_{1},\dotsc ,x_{n}]$:
 \begin{equation*}1=\Res \begin{bmatrix}\Delta _{N,U,V}(x,y)dx_{1}\wedge 
\dotsb \wedge dx_{n}\\
U^{1}(x)^{NB}\scalar {V^{1},P},\dotsc ,U^{n}(x)^{NB}\scalar 
 This formula, which is a polynomial identity in $y$, holds whenever
$\Delta _{N,U,V}(x,y)$ is the determinant of an arbitrary $n\times n$ matrix
whose coefficients $\delta _{jl}\in \K [U,V,x,y]$ have degree in $x,y$ at most
$NB+D_{1}-1$ and for all $j=1,\dotsc ,n$ satisfy the relations
 \begin{equation*}U^{j}(x)^{NB}\scalar {V^{j},P(x)}-U^{j}(y)^{NB}\scalar 
=\sum _{l=1}^{n} (x_{l}-y_{l})\delta _{jl}(x,y).\end{equation*}

\section*{3. Effective Nullstellensatz}In this section, $\A $ will be a 
unitary factorial regular integral domain
with a size $\tb $; its quotient field will be denoted by $\K $ and assumed
to be infinite. The size $\tb $ can be extended to $\Pol (\A )$  with values 
in $\{-\infty \}\cup [0,+\infty [$; we refer to \cite{Ph1} for this purpose.
 In the following theorem we need an auxiliary function defined for
$s\in \N \cup \{+\infty \}$ by
 \begin{equation*}\vartheta (s)=\inf \{\xi \in [0,\infty [\,:\,\vartheta 
_{0}(\xi )\geq s\},\end{equation*}
 \begin{equation*}\vartheta _{0}(\xi )=\#\{a\in \A \,:\,\tb (a)\leq \xi 
\},\quad \xi \in [0,\infty [.\end{equation*}
 Let us state now our final result, that is, our solution of the B\'{e}zout
identity with good degree and size estimates.

  Let $p_{1},\dotsc ,p_{M}\in \A [x_{1},\dotsc ,x_{n}]$ and let $\A $ be an 
domain with infinite quotient field $\K $.  The ring $\A $ is assumed to be a
factorial regular ring with Krull dimension $\kappa $ and equipped with a
size $\tb $ (with corresponding $c,c',\vartheta $).  The degrees $D_{j}=\deg 
p_{j}$ are assumed to be in decreasing order and
$h:=\max \{\tb (p_{j}),c'\log (n+2)\}$.  If $p_{1},\dotsc ,p_{M}$ have no 
common zero
in some algebraic closure $\overline{\K }$ of $\K $, there exists $r_{0}\in 
\A $,
and $q_{1},\dotsc ,q_{N}\in \A [x]$, such that
 \begin{equation*}r_{0}=\sum _{j=1}^{M} q_{j} p_{j},\end{equation*}
 with the estimates
 \begin{equation*}\begin{cases}\deg (p_{j} q_{j})\leq n(n+1)^{3} 
B(D_{1},\dotsc ,D_{n})+n(D_{1}-1),\\
\tb (p_{j}q_{j})\leq C_{0}\varpi ^{4} 2^{n} n^{17}c^{16}B^{4} D_{1}^{2}
\bigl (h+\vartheta [(\gamma _{0} D_{1})^{n}]+c'\log M+c'D_{1}\log (2n+
2)\bigr ),\\
\tb (r_{0})\leq C_{0}\varpi ^{4} 2^{n} n^{17}c^{16}B^{4} D_{1}^{2}
\bigl (h+\vartheta [(\gamma _{0} D_{1})^{n}]+c'\log M+c'D_{1}\log (2n+
2)\bigr ),
 where $\gamma _{0},C_{0}$ are absolute integral constants,
$B=B(D_{1},\dotsc ,D_{n})$ is defined in~Section~2, and the constants $c,c'$
depend only on the size $\tb $, while $\varpi $ depends both on $n$ and
$\tb $.

  In the two examples mentioned earlier, one can make the constants
$c,c',\varpi $ and the function $\vartheta $ explicit.  Namely, for $\A =\Z $,
we have $c=3$, $c'=1$, and $\varpi =9(n+1)2^{n+2}(1+4\log (n+1))^{n+2}$.  For
$\A =\Fb _{p}[\tau _{1},\dotsc ,\tau _{q}]$, we can take $c=1$, $c'=0$, and
$\varpi =2n+q+1$.  Similarly, in the first example, $\vartheta (s)\simeq 
\log s$, and in the second, $\vartheta (s)\simeq (\log s/\log p)^{1/q}$.

\begin{proof}[Sketch of the proof]
  The key idea of the proof is to consider $n$ affine forms (with
coefficents in $\A $), $U_{1},\dotsc ,U_{n}$, and $n$ linear combinations
$\scalar {V^{j},P}$ (with coefficients in $\A $) of the entries $p_{j}$, such
that the hypotheses of~Proposition~7 are fulfilled.  Then, we start with
the Kronecker identity
 \begin{equation*}1=\Res \begin{bmatrix}\Delta _{N,U,V}(x,y)dx_{1}\wedge 
\dotsb \wedge dx_{n}\\
U^{1}(x)^{NB}\scalar {V^{1},P},\dotsc ,U^{n}(x)^{NB}\scalar 
 which is valid for $N=(n+1)^{3}$.  From now on, we assume that the variables
$U,V$ have been fixed and so we will drop them, as well as $N$, from the
notation, when convenient.  In particular, we will denote by
$\Theta _{j}(x)=U^{j}(x)^{NB}\scalar {V^{j},P(x)}$, for $j=1,\dotsc ,n$. 
Correspondingly, we let $\Delta (x,y)=\Delta _{N,U,V}(x,y)$, and
$\delta _{ij}$ be the entries of the corresponding matrix.  We introduce a
linear combination $q$ of the $p_{j}$ such that the ideal
$I(\Theta _{1},\dotsc ,\Theta _{n},q)=\K [x_{1},\dotsc ,x_{n}]$, and 
$g_{n+1,l}$, for $1\leq l\leq n$, in $2n$ variables such that
$q(x)-q(y)=\sum _{l=1}^{n} g_{n+1,l}(x,y)(x_{l}-y_{l}).$
 Thus, we can rewrite the determinant $\Delta $ as
 \begin{equation*}(1/q(x))\vmatrix \delta _{11}&\hdots &\delta _{1n}&g_{n+
\vdots &\ddots &\vdots &\vdots \\
\delta _{n1}&\hdots &\delta _{nn}&g_{n+1,n}(x,y)\\
\Theta _{1}(y)-\Theta _{1}(x)&\hdots &\Theta _{n}(y)-\Theta 
_{n}(x)&q(y)\endvmatrix \end{equation*}
 and develop this new $(n+1)\times (n+1)$ determinant along the last row to
 \begin{equation*}\Delta (x,y)=(1/q(x))\biggl (\biggl (\sum _{j=1}^{n}
\bigl (\Theta _{j}(y)-\Theta _{j}(x)\bigr )\Delta _{j}(x,y)\biggr )
+q(y)\Delta (x,y)\biggr ).\end{equation*}
 Since the residue symbol is annihilated by the ideal, we can
rewrite (7) as a B\'{e}zout identity
 \begin{equation*}1=\sum _{j=1}^{n}\Res \begin{bmatrix}\Delta 
_{j}(x,y)dx\big /q(x)\\
\Theta _{1}(x),\dotsc ,\Theta _{n}(x)\end{bmatrix}
\Theta _{j}(y)
+\Res \begin{bmatrix}\Delta (x,y)dx\big /q(x)\\
\Theta _{1}(x),\dotsc ,\Theta _{n}(x)\end{bmatrix}
 It is clear that this is an identity of the form
$1=\sum _{j=1}^{M} p_{j}(y)q_{j}(y),$
 where the $q_{j}$ are in $\K [x]$.  This is the formula that solves the
 Nullstellensatz with good estimates.  Note that, up to this
point, we already have the estimates for the degrees.  The size estimates
are obtained from the results in Section~1.\end{proof}


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